What system of linear equations is represented by the lines shown?
Choice D is correct. A line in the xy-plane that passes through the points and has slope , where , and can be defined by an equation of the form . One of the lines shown in the graph passes through the points and . Substituting for , for , for , and for in the equation yields , or . Substituting for , for and for in the equation yields , which is equivalent to . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, an equation of this line is . Similarly, the other line shown in the graph passes through the points and . Substituting for , for , for , and for in the equation yields , or . Substituting for , for , and for in the equation yields , which is equivalent to . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, an equation of this line is . So, the system of linear equations represented by the lines shown is and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of when ?
Choice C is correct. It’s given that the function is defined by . Substituting for in this equation yields , which is equivalent to , or . Therefore, the value of is when .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
The correct answer is . It’s given that line is perpendicular to line in the xy-plane. This means that the slope of line is the negative reciprocal of the slope of line . The equation of line , , is written in slope-intercept form , where is the slope of the line and is the y-coordinate of the y-intercept of the line. It follows that the slope of line is . The negative reciprocal of a number is divided by the number. Therefore, the negative reciprocal of is , or . Thus, the slope of line is . Note that 3/17, .1764, .1765, and 0.176 are examples of ways to enter a correct answer.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Adding the second equation to the first equation in the given system of equations yields , or . Dividing both sides of this equation by yields . Substituting for in the first equation, , yields, or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for and for in the expression yields , or . Therefore, the value of is .
A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?
2
4
5
6
Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality represents this situation. It’s also given that the helicopter must carry at least 10 packages: the inequality
represents this situation. Values of a and b that satisfy these two inequalities represent the allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to be minimized. Expressing b in terms of a in the second inequality yields
, so the minimum value of b is equal to
. Substituting
for b in the first inequality results in
. Using the distributive property to rewrite this inequality yields
, or
. Subtracting 1,000 from both sides of this inequality yields
. Dividing both sides of this inequality by 20 results in
. This means that the maximum number of 120-pound packages that the helicopter can carry per trip is 5.
Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.
If , what is the value of ?
The correct answer is . It’s given that . Multiplying each side of this equation by yields , or . Subtracting from each side of this equation yields , or . Therefore, the value of is .
The function is defined by . For what value of does ?
Choice B is correct. Substituting for in the given function yields . Dividing each side of this equation by yields . Therefore, when the value of is .
Choice A is incorrect. This is the value of for which , not .
Choice C is incorrect. This is the value of for which , not .
Choice D is incorrect. This is the value of for which , not .
For the linear function , is a constant and . What is the value of ?
Choice A is correct. For the linear function , it’s given that . Substituting for and for in the given function yields , or . Subtracting from each side of this equation yields . Therefore, the value of is .
Choice B is incorrect. Substituting for in the given function yields . For this function, when the value of is , the value of is , not .
Choice C is incorrect. Substituting for in the given function yields . For this function, when the value of is , the value of is , not .
Choice D is incorrect. Substituting for in the given function yields . For this function, when the value of is , the value of is , not .
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is , where and are constants. What is the value of ?
The correct answer is . It’s given in the table that the coordinates of two points on a line in the xy-plane are and . The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, , between any two points, and , on the line can be calculated using the slope formula, . It follows that the slope of the line with the given points from the table, and , is , which is equivalent to , or . It's given that the y-intercept of the line is . Substituting for and the coordinates of the points and into the slope formula yields , which is equivalent to , or . Multiplying both sides of this equation by yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the value of is .
Gabriella deposits in a savings account at the end of each week. At the beginning of the week of a year there was in that savings account. How much money, in dollars, will be in the account at the end of the week of that year?
Choice D is correct. It’s given that at the beginning of the week of the year there was in a savings account and Gabriella deposits in that savings account at the end of each week. Therefore, the amount of money, in dollars, in the savings account at the end of the week of that year is , or .
Choice A is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the week if Gabriella withdraws, rather than deposits, at the end of each week.
Choice B is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the week, not the week.
Choice C is incorrect and may result from conceptual or calculation errors.
One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?
Choice B is correct. A system of two linear equations in two variables, and , has no solution when the lines in the xy-plane representing the equations are parallel and distinct. Two lines are parallel and distinct if their slopes are the same and their y-intercepts are different. The slope of the graph of the given equation, , in the xy-plane can be found by rewriting the equation in the form , where is the slope of the graph and is the y-intercept. Adding to each side of the given equation yields . Dividing each side of this equation by yields , or . It follows that the slope of the graph of the given equation is and the y-intercept is . Therefore, the graph of the second equation in the system must also have a slope of , but must not have a y-intercept of . Multiplying each side of the equation given in choice B by yields , or . It follows that the graph representing the equation in choice B has a slope of and a y-intercept of . Since the slopes of the graphs of the two equations are equal and the y-intercepts of the graphs of the two equations are different, the equation in choice B could be the second equation in the system.
Choice A is incorrect. This equation can be rewritten as . It follows that the graph of this equation has a slope of , so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.
Choice C is incorrect. This equation can be rewritten as . It follows that the graph of this equation has a slope of and a y-intercept of , so the system consisting of this equation and the given equation has infinitely many solutions, rather than no solution.
Choice D is incorrect. This equation can be rewritten as . It follows that the graph of this equation has a slope of , so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.
If satisfies the system of equations above, what is the value of y ?
The correct answer is . One method for solving the system of equations for y is to add corresponding sides of the two equations. Adding the left-hand sides gives
, or 4y. Adding the right-hand sides yields
. It follows that
. Finally, dividing both sides of
by 4 yields
or
. Note that 3/2 and 1.5 are examples of ways to enter a correct answer.
The system of equations above has solution (x, y). What is the value of x ?
3
4
6
Choice D is correct. Adding the corresponding sides of the two equations eliminates y and yields , as shown.
If (x, y) is a solution to the system, then (x, y) satisfies both equations in the system and any equation derived from them. Therefore, .
Choices A, B, and C are incorrect and may be the result of errors when solving the system.
A veterinarian recommends that each day a certain rabbit should eat calories per pound of the rabbit’s weight, plus an additional calories. Which equation represents this situation, where is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is pounds?
Choice D is correct. It’s given that a veterinarian recommends that each day the rabbit should eat calories per pound of the rabbit’s weight, plus an additional calories. If the rabbit’s weight is pounds, then multiplying calories per pound by the rabbit’s weight, pounds, yields calories. Adding the additional calories that the rabbit should eat each day yields calories. It’s given that is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is pounds. Therefore, this situation can be represented by the equation .
Choice A is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat calories per pound of the rabbit’s weight.
Choice B is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat , or , calories per pound of the rabbit’s weight.
Choice C is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat calories per pound of the rabbit’s weight, plus an additional calories.
The total cost , in dollars, to lease a car for months from a particular car dealership is given by , where is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is ?
Choice C is correct. It's given that is the total cost, in dollars, to lease a car from this dealership with a monthly payment of dollars. Therefore, the total cost, in dollars, to lease the car when the monthly payment is is represented by the value of when . Substituting for in the equation yields , or . Thus, when the monthly payment is , the total cost to lease a car is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph of the equation is a line in the xy-plane, where a and k are constants. If the line contains the points
and
, what is the value of k ?
2
3
Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, , where m is the slope and b is the y-coordinate of the y-intercept. The equation
can be rewritten in the form
. One of the given points,
, is the y-intercept. Thus, the y-coordinate of the y-intercept
must be equal to
. Multiplying both sides by k gives
. Dividing both sides by
gives
.
Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
The correct answer is . For an equation in slope-intercept form , represents the slope of the line in the xy-plane defined by this equation. It's given that line is defined by . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields , or . Thus, the slope of line in the xy-plane is . Since line is perpendicular to line in the xy-plane, the slope of line is the negative reciprocal of the slope of line . The negative reciprocal of is . Note that 1/4 and .25 are examples of ways to enter a correct answer.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, and , are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients in the other equation, meaning ; and the lines are distinct if the constants are not proportional, meaning is not equal to or . The first equation in the given system is . Multiplying each side of this equation by yields . Adding to each side of this equation yields , or . The second equation in the given system is . Multiplying each side of this equation by yields . Subtracting from each side of this equation yields . Subtracting from each side of this equation yields . Therefore, the two equations in the given system, written in standard form, are and. As previously stated, if this system has no solution, the lines represented by the equations in the xy-plane are parallel and distinct, meaning the proportion , or , is true and the proportion is not true. The proportion is not true. Multiplying each side of the true proportion, , by yields . Therefore, if the system has no solution, then the value of is .
A window repair specialist charges for the first two hours of repair plus an hourly fee for each additional hour. The total cost for hours of repair is . Which function gives the total cost, in dollars, for hours of repair, where ?
Choice A is correct. It’s given that the window repair specialist charges for the first two hours of repair plus an hourly fee for each additional hour. Let represent the hourly fee for each additional hour after the first two hours. Since it’s given that is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression . Therefore, the total cost, in dollars, for hours of repair is . It’s given that the total cost for hours of repair is . Substituting for and for into the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields , which is equivalent to , or . Therefore, the total cost, in dollars, for hours of repair is .
Choice B is incorrect. This function represents the total cost, in dollars, for hours of repair where the specialist charges , rather than , for the first two hours of repair.
Choice C is incorrect. This function represents the total cost, in dollars, for hours of repair where the specialist charges , rather than , for the first two hours of repair, and an hourly fee of , rather than , after the first two hours.
Choice D is incorrect. This function represents the total cost, in dollars, for hours of repair where the specialist charges , rather than , for the first two hours of repair, and an hourly fee of , rather than , after the first two hours.
Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin contained 24,000 bushels of corn when Hector began to use the auger. After 5 hours of using the auger, 19,350 bushels of corn remained in the bin. If the auger continues to remove corn at this rate, what is the total number of hours Hector will have been using the auger when 12,840 bushels of corn remain in the bin?
3
7
8
12
Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During the 5-hour period, the auger removed corn from the bin at a constant rate of bushels per hour. Assuming the auger continues to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12 hours when 12,840 bushels of corn remain in the storage bin.
Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840.
The function is defined by . The graph of in the xy-plane has an x-intercept at and a y-intercept at , where and are constants. What is the value of ?
Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where . It's given that function is defined by . Therefore, the equation representing the graph of is . Substituting for in the equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the x-intercept of the graph of in the xy-plane is . It's given that the x-intercept of the graph of is . Therefore, . The y-intercept of a graph in the xy-plane is the point on the graph where . Substituting for in the equation yields , or . Therefore, the y-intercept of the graph of in the xy-plane is . It's given that the y-intercept of the graph of is . Therefore, . If and , then the value of is , or .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of , not .
If , what is the value of ?
Choice C is correct. Dividing all terms in the given equation by yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For line , the table shows three values of and their corresponding values of . Line is the result of translating line down units in the xy-plane. What is the x-intercept of line ?
Choice D is correct. The equation of line can be written in slope-intercept form , where is the slope of the line and is the y-intercept of the line. It’s given that line contains the points , , and . Therefore, its slope can be found as , or . Substituting for in the equation yields . Substituting for and for in this equation yields , or . Subtracting from both sides of this equation yields . Substituting for in yields . Since line is the result of translating line down units, an equation of line is , or . Substituting for in this equation yields . Solving this equation for yields . Therefore, the x-intercept of line is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?
16,250
16,500
16,750
17,500
Choice A is correct. Let the economist’s model be the linear function , where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where
. Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs
, two points that satisfy the function are
and
. The slope m of the function can be found using the formula
. Substituting the given values into this formula yields
, or
. Therefore,
. The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point
yields
, or
. Adding 10,000 to both sides of this equation yields
. Therefore, the linear function the economist used as the model is
. Substituting 55 for P yields
. It follows that when the selling price is $55 per unit, the demand is 16,250 units.
Choices B, C, and D are incorrect and may result from calculation or conceptual errors.
A certain apprentice has enrolled in hours of training courses. The equation represents this situation, where is the number of on-site training courses and is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?
The correct answer is . It's given that the equation represents the situation, where is the number of on-site training courses, is the number of online training courses, and is the total number of hours of training courses the apprentice has enrolled in. Therefore, represents the number of hours the apprentice has enrolled in on-site training courses, and represents the number of hours the apprentice has enrolled in online training courses. Since is the number of on-site training courses and is the number of online training courses the apprentice has enrolled in, is the number of hours each on-site course takes and is the number of hours each online course takes. Subtracting these numbers gives , or more hours each online training course takes than each on-site training course.
Hiro and Sofia purchased shirts and pants from a store. The price of each shirt purchased was the same and the price of each pair of pants purchased was the same. Hiro purchased 4 shirts and 2 pairs of pants for $86, and Sofia purchased 3 shirts and 5 pairs of pants for $166. Which of the following systems of linear equations represents the situation, if x represents the price, in dollars, of each shirt and y represents the price, in dollars, of each pair of pants?
Choice A is correct. Hiro purchased 4 shirts and each shirt cost x dollars, so he spent a total of 4x dollars on shirts. Likewise, Hiro purchased 2 pairs of pants, and each pair of pants cost y dollars, so he spent a total of 2y dollars on pants. Therefore, the total amount that Hiro spent was 4x + 2y. Since Hiro spent $86 in total, this can be modeled by the equation 4x + 2y = 86. Using the same reasoning, Sofia bought 3 shirts at x dollars each and 5 pairs of pants at y dollars each, so she spent a total of 3x + 5y dollars on shirts and pants. Since Sofia spent $166 in total, this can be modeled by the equation 3x + 5y = 166.
Choice B is incorrect and may be the result of switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased. Choice C is incorrect and may be the result of switching the total price each person paid. Choice D is incorrect and may be the result of switching the total price each person paid as well as switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased.
The equation shown gives the estimated amount of diesel , in gallons, that remains in the gas tank of a truck after being driven miles, where . What is the estimated amount of diesel, in gallons, that remains in the gas tank of the truck when ?
Choice B is correct. It’s given that the equation gives the estimated amount of diesel , in gallons, that remains in the gas tank of the truck after being driven miles. Substituting for in the given equation yields , which is equivalent to , or . Therefore, the estimated amount of diesel that remains in the gas tank of the truck when is gallons.
Choice A is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when , not when .
Choice C is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when , not when .
Choice D is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when , not when .
In the given system of equations, and are constants. The graphs of these equations in the xy-plane intersect at the point . What is the value of ?
Choice D is correct. It’s given that the graphs of the given system of equations intersect at the point . Therefore, is the solution to the given system. Multiplying the first equation in the given system by yields . Adding this equation to the second equation in the system yields , or . Since is the solution to the system, the value of can be found by substituting for in this equation, which yields . Dividing both sides of this equation by yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A store sells two different-sized containers of a certain Greek yogurt. The store’s sales of this Greek yogurt totaled dollars last month. The equation represents this situation, where is the number of smaller containers sold and is the number of larger containers sold. According to the equation, which of the following represents the price, in dollars, of each smaller container?
Choice A is correct. It's given that the store's sales of a certain Greek yogurt totaled dollars last month. It's also given that the equation represents this situation, where is the number of smaller containers sold and is the number of larger containers sold. Since represents the number of smaller containers of yogurt sold, the expression represents the total sales, in dollars, from smaller containers of yogurt. This means that smaller containers of yogurt were sold at a price of dollars each. Therefore, according to the equation, represents the price, in dollars, of each smaller container.
Choice B is incorrect. This expression represents the total sales, in dollars, from selling larger containers of yogurt.
Choice C is incorrect. This value represents the price, in dollars, of each larger container of yogurt.
Choice D is incorrect. This expression represents the total sales, in dollars, from selling smaller containers of yogurt.
Which equation has the same solution as the given equation?
Choice A is correct. Dividing each side of the given equation by yields , or . Therefore, the equation is equivalent to the given equation and has the same solution.
Choice B is incorrect. This equation is equivalent to , not .
Choice C is incorrect. Distributing on the left-hand side of the given equation yields , not .
Choice D is incorrect. Distributing on the left-hand side of the given equation yields , not .
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
Choice A is correct. It's given that line is perpendicular to line in the xy-plane. It follows that the slope of line is the opposite reciprocal of the slope of line . The equation for line is written in slope-intercept form , where is the slope of the line and is the y-coordinate of the y-intercept of the line. It follows that the slope of line is . The opposite reciprocal of a number is divided by the number. Thus, the opposite reciprocal of is . Therefore, the slope of line is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The point in the xy-plane is a solution to which of the following systems of inequalities?
Choice A is correct. The given point, , is located in the first quadrant in the xy-plane. The system of inequalities in choice A represents all the points in the first quadrant in the xy-plane. Therefore, is a solution to the system of inequalities in choice A.
Alternate approach: Substituting for in the first inequality in choice A, , yields , which is true. Substituting for in the second inequality in choice A, , yields , which is true. Since the coordinates of the point make the inequalities and true, the point is a solution to the system of inequalities consisting of and .
Choice B is incorrect. This system of inequalities represents all the points in the fourth quadrant, not the first quadrant, in the xy-plane.
Choice C is incorrect. This system of inequalities represents all the points in the second quadrant, not the first quadrant, in the xy-plane.
Choice D is incorrect. This system of inequalities represents all the points in the third quadrant, not the first quadrant, in the xy-plane.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. Adding the second equation of the given system to the first equation yields , which is equivalent to . So the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of .
Choice D is incorrect and may result from conceptual or calculation errors.
The cost of renting a backhoe for up to days is for the first day and for each additional day. Which of the following equations gives the cost , in dollars, of renting the backhoe for days, where is a positive integer and ?
Choice D is correct. It's given that the cost of renting a backhoe for up to days is for the first day and for each additional day. Therefore, the cost , in dollars, for days, where , is the sum of the cost for the first day, , and the cost for the additional days, . It follows that , which is equivalent to , or .
Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is for the first day and for each additional day.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
What value of satisfies the equation ?
Choice A is correct. Subtracting from both sides of the given equation yields . Dividing both sides of this equation by yields . Therefore, the value of that satisfies the equation is .
Choice B is incorrect. This value of satisfies the equation .
Choice C is incorrect. This value of satisfies the equation .
Choice D is incorrect. This value of satisfies the equation .
The front of a roller-coaster car is at the bottom of a hill and is 15 feet above the ground. If the front of the roller-coaster car rises at a constant rate of 8 feet per second, which of the following equations gives the height h, in feet, of the front of the roller-coaster car s seconds after it starts up the hill?
Choice A is correct. It’s given that the front of the roller-coaster car starts rising when it’s 15 feet above the ground. This initial height of 15 feet can be represented by a constant term, 15, in an equation. Each second, the front of the roller-coaster car rises 8 feet, which can be represented by 8s. Thus, the equation gives the height, in feet, of the front of the roller-coaster car s seconds after it starts up the hill.
Choices B and C are incorrect and may result from conceptual errors in creating a linear equation. Choice D is incorrect and may result from switching the rate at which the roller-coaster car rises with its initial height.
According to a model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to four times the body weight of the bee, in grams. According to the model, what would be the head width, in millimeters, of a worker bumblebee that has a body weight of 0.5 grams?
The correct answer is 2.6. According to the model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to 4 times the body weight, in grams, of the bee. Let x represent the body weight, in grams, of a worker bumblebee and let y represent the head width, in millimeters. Translating the verbal description of the model into an equation yields . Substituting 0.5 grams for x in this equation yields
, or
. Therefore, a worker bumblebee with a body weight of 0.5 grams has an estimated head width of 2.6 millimeters. Note that 2.6 and 13/5 are examples of ways to enter a correct answer.
The graph of a system of linear equations is shown. What is the solution to the system?
Choice C is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when and , or at the point . Therefore, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
The correct answer is . For an equation in slope-intercept form , represents the slope of the line in the xy-plane defined by this equation. It's given that line is defined by . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields , or . Thus, the slope of line is . If line is perpendicular to line , then the slope of line is the negative reciprocal of the slope of line . The negative reciprocal of is . Note that 1/2 and .5 are examples of ways to enter a correct answer.
If , what is the value of ?
Choice B is correct. Dividing each side of the given equation by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
Choice C is correct. It’s given that line is perpendicular to line in the xy-plane. This means that the slope of line is the negative reciprocal of the slope of line . If the equation for line is rewritten in slope-intercept form , where and are constants, then is the slope of the line and is its y-intercept. Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . It follows that the slope of line is . The negative reciprocal of a number is divided by the number. Therefore, the negative reciprocal of is , or . Thus, the slope of line is .
Choice A is incorrect. This is the slope of line , not line .
Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line .
Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line .
What value of is the solution to the given equation?
The correct answer is . It’s given that . Combining like terms on the left-hand side of this equation yields . Subtracting from each side of this equation yields . Therefore, the value of that's the solution to the given equation is .
Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school?
Choice D is correct. It is given that w is the number of minutes that Adam waits for the bus. The total time it takes Adam to get to school on a day he takes the bus is the sum of the minutes, w, he waits for the bus and the 5 minutes the bus ride takes; thus, this time, in minutes, is w + 5. It is also given that the total amount of time it takes Adam to get to school on a day that he walks is 20 minutes. Therefore, w + 5 > 20 gives the values of w for which it would be faster for Adam to walk to school.
Choices A and B are incorrect because w – 5 is not the total length of time for Adam to wait for and then take the bus to school. Choice C is incorrect because the inequality should be true when walking 20 minutes is faster than the time it takes Adam to wait for and ride the bus, not less.
If f is the function defined by , what is the value of
?
3
9
Choice C is correct. If , then
.
Choice A is incorrect and may result from not multiplying x by 2 in the numerator. Choice B is incorrect and may result from dividing 2x by 3 and then subtracting 1. Choice D is incorrect and may result from evaluating only the numerator 2x – 1.
What value of is the solution to the given equation?
Choice B is correct. Subtracting from each side of the given equation yields . Therefore, the value of that is the solution to the given equation is .
Choice A is incorrect. This is the value of that is the solution to the equation , not .
Choice C is incorrect. This is the value of that is the solution to the equation , not .
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by , where is a constant, and . What is the value of ?
The correct answer is . It’s given that . Therefore, for the given function , when , . Substituting for and for in the given function yields , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Substituting for in the given function yields , which is equivalent to , or . Substituting for into this equation yields , or . Therefore, the value of is .
In the given system of equations, and are constants. The system has infinitely many solutions. What is the value of ?
The correct answer is . It’s given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions if and only if the two linear equations are equivalent. Multiplying each side of the first equation in the system by yields , or . Since this equation is equivalent to the second equation and has the same right side as the second equation, the coefficients of and , respectively, should also be the same. It follows that and . Therefore, the value of is , or . Note that 2/7, .2857, 0.285, and 0.286 are examples of ways to enter a correct answer.
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice D is correct. All the tables in the choices have the same three values of , , , and , so each of the three values of can be substituted in the given inequality to compare the corresponding values of in each of the tables. Substituting for in the given inequality yields , or . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields . Therefore, when , the corresponding value of must be less than . Substituting for in the given inequality yields , or . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields . Therefore, when , the corresponding value of must be less than . Substituting for in the given inequality yields , or . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields . Therefore, when , the corresponding value of must be less than . For the table in choice D, when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than . Therefore, the table in choice D gives values of and their corresponding values of that are all solutions to the given inequality.
Choice A is incorrect. When , the corresponding value of in this table is , which isn't less than .
Choice B is incorrect. When , the corresponding value of in this table is , which isn't less than .
Choice C is incorrect. When , the corresponding value of in this table is , which isn't less than .
The given equation represents the distance , in inches, where represents the number of seconds since an object started moving. Which of the following is the best interpretation of in this context?
The object moved a total of inches.
The object moved a total of inches.
The object is moving at a rate of inches per second.
The object is moving at a rate of inches per second.
Choice C is correct. It’s given that in the equation , represents the distance, in inches, and represents the number of seconds since an object started moving. In this equation, is being multiplied by . This means that the object’s distance increases by inches each second. Therefore, the best interpretation of in this context is that the object is moving at a rate of inches per second.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect. This is the best interpretation of , rather than , in this context.
Choice D is incorrect and may result from conceptual errors.
For each real number , which of the following points lies on the graph of each equation in the xy-plane for the given system?
Choice D is correct. Dividing each side of the second equation in the given system by yields . It follows that the two equations in the given system are equivalent and any point that lies on the graph of one equation will also lie on the graph of the other equation. Substituting for in the equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, the point lies on the graph of each equation in the xy-plane for each real number .
Choice A is incorrect. Substituting for in the equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, the point , not the point , lies on the graph of each equation.
Choice B is incorrect. Substituting for in the equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, the point , not the point , lies on the graph of each equation.
Choice C is incorrect. Substituting for in the equation yields , or . Subtracting from each side of this equation yields , or . Dividing each side of this equation by yields . Therefore, the point , not the point , lies on the graph of each equation.
What value of is the solution to the given equation?
Choice A is correct. Subtracting from both sides of the given equation yields . Subtracting from both sides of this equation yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For the given linear function , which table gives three values of and their corresponding values of ?
Choice B is correct. For the given linear function , must equal for all values of . Of the given choices, only choice B gives three values of and their corresponding values of for the given linear function .
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
A mixture consisting of only vitamin D and calcium has a total mass of grams. The mass of vitamin D in the mixture is grams. What is the mass, in grams, of calcium in the mixture?
Choice C is correct. Let represent the mass, in grams, of vitamin D in the mixture, and let represent the mass, in grams, of calcium in the mixture. It’s given that the mixture consists of only vitamin D and calcium and that the total mass of the mixture is grams. Therefore, the equation represents this situation. It’s also given that the mass of vitamin D in the mixture is grams. Substituting for in the equation yields . Subtracting from both sides of this equation yields . Therefore, the mass of calcium in the mixture is grams.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the total mass, in grams, of the mixture, not the mass, in grams, of calcium in the mixture.
Choice D is incorrect. This is the mass, in grams, of vitamin D in the mixture, not the mass, in grams, of calcium in the mixture.
The function is defined by . For what value of is ?
The correct answer is . It’s given that . Substituting for in the given function yields . Dividing both sides of this equation by yields . Therefore, the value of when is .
What is the solution to the given equation?
The correct answer is . Subtracting from both sides of the given equation yields . Dividing both sides of this equation by yields . Therefore, the solution to the given equation is .
The graph of the linear function is shown, where . What is the x-intercept of the graph of ?
Choice A is correct. The x-intercept of a graph is the point where the graph intersects the x-axis. The graph of function , where , intersects the x-axis at . Therefore, the x-intercept of the graph of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A company that provides whale-watching tours takes groups of people at a time. The company’s revenue is dollars per adult and dollars per child. If the company’s revenue for one group consisting of adults and children was dollars, how many people in the group were children?
Choice C is correct. Let represent the number of children in a whale-watching tour group. Let represent the number of adults in this group. Because it's given that people are in a group and the group consists of adults and children, it must be true that . Since the company's revenue is dollars per child, the total revenue from children in this group was dollars. Since the company's revenue is dollars per adult, the total revenue from adults in this group was dollars. Because it's given that the total revenue for this group was dollars, it must be true that . The equations and form a linear system of equations that can be solved to find the value of , which represents the number of children in the group, using the elimination method. Multiplying both sides of the equation by yields . Subtracting from yields , which is equivalent to , or . Dividing both sides of this equation by yields . Therefore, people in the group were children.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the number of adults in the group, not the number of children in the group.
Choice D is incorrect and may result from conceptual or calculation errors.
Line is defined by . Line is parallel to line in the xy-plane. What is the slope of line ?
Choice B is correct. It's given that line is defined by . For an equation of a line written in the form , is the slope of the line and is the y-coordinate of the y-intercept of the line. It follows that the slope of line is . It's also given that line is parallel to line in the xy-plane. Since parallel lines have equal slopes, line also has a slope of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the y-coordinate of the y-intercept of line , not the slope of line .
Choice D is incorrect and may result from conceptual or calculation errors.
How many solutions does the given system of equations have?
Exactly one
Exactly two
Infinitely many
Zero
Choice D is correct. A system of two linear equations in two variables, and , has zero solutions if the lines representing the equations in the xy-plane are distinct and parallel. Two lines are distinct and parallel if they have the same slope but different y-intercepts. Each equation in the given system can be written in slope-intercept form , where is the slope of the line representing the equation in the xy-plane and is the y-intercept. Adding to both sides of the first equation in the given system of equations, , yields . Dividing both sides of this equation by yields . It follows that the first equation in the given system of equations has a slope of and a y-intercept of . Adding to both sides of the second equation in the given system of equations, , yields . Dividing both sides of this equation by yields . It follows that the second equation in the given system of equations has a slope of and a y-intercept of . Since the slopes of these lines are the same and the y-intercepts are different, it follows that the given system of equations has zero solutions.
Alternate approach: To solve the system by elimination, multiplying the second equation in the given system of equations, , by yields . Adding this equation to the first equation in the given system of equations, , yields , or . Since this equation isn't true, the given system of equations has zero solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
In August, a car dealer completed more than times the number of sales the car dealer completed in September. In August and September, the car dealer completed sales. How many sales did the car dealer complete in September?
The correct answer is . It’s given that in August, the car dealer completed more than times the number of sales the car dealer completed in September. Let represent the number of sales the car dealer completed in September. It follows that represents the number of sales the car dealer completed in August. It’s also given that in August and September, the car dealer completed sales. It follows that , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, the car dealer completed sales in September.
In the xy-plane, the graph of the linear function contains the points and . Which equation defines , where ?
Choice C is correct. In the xy-plane, an equation of the graph of a linear function can be written in the form , where represents the slope and represents the y-intercept of the graph of . It’s given that the graph of the linear function , where , in the xy-plane contains the point . Thus, . The slope of the graph of a line containing any two points and can be found using the slope formula, . Since it’s given that the graph of the linear function contains the points and , it follows that the slope of the graph of the line containing these points is , or . Substituting for and for in yields .
Choice A is incorrect. This function represents a graph with a slope of and a y-intercept of .
Choice B is incorrect. This function represents a graph with a slope of and a y-intercept of .
Choice D is incorrect. This function represents a graph with a slope of and a y-intercept of .
What is the solution to the given system of equations?
Choice A is correct. The second equation in the given system is . Substituting for in the first equation in the given system yields . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the solution to the given system of equations is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the solution , not , to the given system of equations.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph of the linear function is shown. What is the y-intercept of the graph of ?
Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of shown intersects the y-axis at the point . Therefore, the y-intercept of the graph of is .
Choice A is incorrect. This is the x-intercept, not the y-intercept, of the graph of .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The number is less than the number . Which equation represents the relationship between and ?
Choice D is correct. It’s given that the number is less than the number . A number that's less than the number is equivalent to subtracted from the number , or . Therefore, the equation represents the relationship between and .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The function is defined by . What is the value of ?
Choice B is correct. It’s given that the function is defined by . Substituting for in the given function yields , which is equivalent to , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of for the function , not .
Choice D is incorrect. This is the value of for the function , not .
The function is defined by the equation . What is the value of when ?
The correct answer is . The value of when can be found by substituting for in the given equation . This yields , or . Therefore, when , the value of is .
At how many points do the graphs of the equations and intersect in the xy-plane?
Choice B is correct. Each given equation is written in slope-intercept form, , where is the slope and is the y-intercept of the graph of the equation in the xy-plane. The graphs of two lines that have different slopes will intersect at exactly one point. The graph of the first equation is a line with slope . The graph of the second equation is a line with slope . Since the graphs are lines with different slopes, they will intersect at exactly one point.
Choice A is incorrect because two graphs of linear equations have intersection points only if they are parallel and therefore have the same slope.
Choice C is incorrect because two graphs of linear equations in the xy-plane can have only , , or infinitely many points of intersection.
Choice D is incorrect because two graphs of linear equations in the xy-plane can have only , , or infinitely many points of intersection.
Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if x is the number of junior directors and y is the number of senior directors?
Choice B is correct. Marisa will hire x junior directors and y senior directors. Since she needs to hire at least 10 staff members, . Each junior director will be paid $640 per week, and each senior director will be paid $880 per week. Marisa’s budget for paying the new staff is no more than $9,700 per week; in terms of x and y, this condition is
. Since Marisa must hire at least 3 junior directors and at least 1 senior director, it follows that
and
. All four of these conditions are represented correctly in choice B.
Choices A and C are incorrect. For example, the first condition, , in each of these options implies that Marisa can pay the new staff members more than her budget of $9,700. Choice D is incorrect because Marisa needs to hire at least 10 staff members, not at most 10 staff members, as the inequality
implies.
For a snowstorm in a certain town, the minimum rate of snowfall recorded was inches per hour, and the maximum rate of snowfall recorded was inches per hour. Which inequality is true for all values of , where represents a rate of snowfall, in inches per hour, recorded for this snowstorm?
Choice D is correct. It's given that for a snowstorm in a certain town, the minimum rate of snowfall recorded was inches per hour, the maximum rate of snowfall recorded was inches per hour, and represents a rate of snowfall, in inches per hour, recorded for this snowstorm. It follows that the inequality is true for all values of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
During a portion of a flight, a small airplane's cruising speed varied between miles per hour and miles per hour. Which inequality best represents this situation, where is the cruising speed, in miles per hour, during this portion of the flight?
Choice D is correct. It's given that during a portion of a flight, a small airplane's cruising speed varied between miles per hour and miles per hour. It's also given that represents the cruising speed, in miles per hour, during this portion of the flight. It follows that the airplane's cruising speed, in miles per hour, was at least , which means , and was at most , which means . Therefore, the inequality that best represents this situation is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
John paid a total of for a microscope by making a down payment of plus monthly payments of each. Which of the following equations represents this situation?
Choice C is correct. It’s given that John made a payment each month for months. The total amount of these payments can be represented by the expression . The down payment can be added to that amount to find the total amount John paid, yielding the expression . It’s given that John paid a total of . Therefore, the expression for the total amount John paid can be set equal to that amount, yielding the equation .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. It's given by the first equation in the system that . Substituting for in the equation yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Valentina bought two containers of beads. In the first container 30% of the beads are red, and in the second container 70% of the beads are red. Together, the containers have at least 400 red beads. Which inequality shows this relationship, where x is the total number of beads in the first container and y is the total number of beads in the second container?
Choice A is correct. It is given that x is the total number of beads in the first container and that 30% of those beads are red; therefore, the expression 0.3x represents the number of red beads in the first container. It is given that y is the total number of beads in the second container and that 70% of those beads are red; therefore, the expression 0.7y represents the number of red beads in the second container. It is also given that, together, the containers have at least 400 red beads, so the inequality that shows this relationship is 0.3x + 0.7y ≥ 400.
Choice B is incorrect because it represents the containers having a total of at most, rather than at least, 400 red beads. Choice C is incorrect and may be the result of misunderstanding how to represent a percentage of beads in each container. Also, the inequality shows the containers having a combined total of at most, rather than at least, 400 red beads. Choice D is incorrect because the percentages were not converted to decimals.
Which table gives three values of and their corresponding values of for the given equation?
Choice A is correct. Substituting for into the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for into the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for into the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Of the choices given, only the table in choice A gives these three values of and their corresponding values of for the given equation.
Choice B is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice C is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice D is incorrect. This table gives three values of and their corresponding values of for the equation .
As part of a science project on evaporation, Amaya measured the height of a liquid in a container over a period of time. The function gives the estimated height, in centimeters (cm), of the liquid in the container days after the start of the project. Which of the following is the best interpretation of in this context?
The estimated height, in cm, of the liquid at the start of the project
The estimated height, in cm, of the liquid at the end of the project
The estimated change in the height, in cm, of the liquid each day
The estimated number of days for all of the liquid to evaporate
Choice A is correct. It's given that the function gives the estimated height, in centimeters (cm), of the liquid in the container days after the start of the project. For a linear function in the form , where and are constants, represents the value of and represents the rate of change of the function. It follows that in the given function, represents the value of . Therefore, the best interpretation of in this context is the estimated height, in cm, of the liquid at the start of the project.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. The estimated change in the height, in cm, of the liquid each day is , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Lorenzo purchased a box of cereal and some strawberries at the grocery store. Lorenzo paid for the box of cereal and per pound for the strawberries. If Lorenzo paid a total of for the box of cereal and the strawberries, which of the following equations can be used to find , the number of pounds of strawberries Lorenzo purchased? (Assume there is no sales tax.)
Choice A is correct. It's given that represents the number of pounds of strawberries Lorenzo purchased and Lorenzo paid per pound for the strawberries. It follows that the total amount, in dollars, Lorenzo paid for strawberries can be represented by . It’s given that Lorenzo paid for the box of cereal. If Lorenzo paid a total of for the box of cereal and strawberries, it follows that the equation can be used to find .
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form , where , , and are constants, are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients for and in the other equation. The first equation in the given system, , can be written in the form by subtracting from both sides of the equation to yield . The second equation in the given system, , can be written in the form by adding to both sides of the equation to yield . The coefficient of in the second equation is times the coefficient of in the first equation. That is, . For the lines to be parallel, the coefficient of in the second equation must also be times the coefficient of in the first equation. Therefore, , or . Thus, if the given system has no solution, the value of is .
The total cost, in dollars, to rent a surfboard consists of a service fee and a per hour rental fee. A person rents a surfboard for hours and intends to spend a maximum of to rent the surfboard. Which inequality represents this situation?
Choice D is correct. The cost of the rental fee depends on the number of hours the surfboard is rented. Multiplying hours by dollars per hour yields a rental fee of dollars. The total cost of the rental consists of the rental fee plus the dollar service fee, which yields a total cost of dollars. Since the person intends to spend a maximum of dollars to rent the surfboard, the total cost must be at most dollars. Therefore, the inequality represents this situation.
Choice A is incorrect. This represents a situation where the rental fee, not the total cost, is at most dollars.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A model predicts that a certain animal weighed pounds when it was born and that the animal gained pounds per day in its first year of life. This model is defined by an equation in the form , where is the predicted weight, in pounds, of the animal days after it was born, and and are constants. What is the value of ?
The correct answer is . For a certain animal, it's given that a model predicts the animal weighed pounds when it was born and gained pounds per day in its first year of life. It's also given that this model is defined by an equation in the form , where is the predicted weight, in pounds, of the animal days after it was born, and and are constants. It follows that represents the predicted weight, in pounds, of the animal when it was born and represents the predicted rate of weight gain, in pounds per day, in its first year of life. Thus, the value of is .
A shipping company charged a customer $25 to ship some small boxes and some large boxes. The equation above represents the relationship between a, the number of small boxes, and b, the number of large boxes, the customer had shipped. If the customer had 3 small boxes shipped, how many large boxes were shipped?
3
4
5
6
Choice B is correct. It’s given that a represents the number of small boxes and b represents the number of large boxes the customer had shipped. If the customer had 3 small boxes shipped, then . Substituting 3 for a in the equation
yields
or
. Subtracting 9 from both sides of the equation yields
. Dividing both sides of this equation by 4 yields
. Therefore, the customer had 4 large boxes shipped.
Choices A, C, and D are incorrect. If the number of large boxes shipped is 3, then . Substituting 3 for b in the given equation yields
or
. Subtracting 12 from both sides of the equation and then dividing by 3 yields
. However, it’s given that the number of small boxes shipped, a, is 3, not
, so b cannot equal 3. Similarly, if
or
, then
or
, respectively, which is also not true.
What is the equation of the line that passes through the point and is parallel to the graph of in the xy-plane?
Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form , where is the slope of the line and is its y-intercept. It’s given that the line passes through the point . Therefore, . It’s also given that the line is parallel to the graph of , which means the line has the same slope as the graph of . The slope of the graph of is . Therefore, . Substituting for and for in the equation yields .
Choice A is incorrect. The graph of this equation passes through the point , not , and has a slope of , not .
Choice C is incorrect. The graph of this equation passes through the point , not .
Choice D is incorrect. The graph of this equation passes through the point , not , and has a slope of , not .
A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents?
30
20
19
18
Choice C is correct. Let x represent the number of 2-person tents and let y represent the number of 4-person tents. It is given that the total number of tents was 60 and the total number of people in the group was 202. This situation can be expressed as a system of two equations, and
. The first equation can be rewritten as
. Substituting
for y in the equation
yields
. Distributing and combining like terms gives
. Subtracting 240 from both sides of
and then dividing both sides by
gives
. Therefore, the number of 2-person tents is 19.
Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240. However, the actual number of people who slept in tents was 202. The difference of 38 accounts for the 2-person tents. Since each of these tents holds 2 people fewer than a 4-person tent, gives the number of 2-person tents.
Choice A is incorrect. This choice may result from assuming exactly half of the tents hold 2 people. If that were true, then the total number of people who slept in tents would be ; however, the total number of people who slept in tents was 202, not 180. Choice B is incorrect. If 20 tents were 2-person tents, then the remaining 40 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be
; however, the total number of people who slept in tents was 202, not 200. Choice D is incorrect. If 18 tents were 2-person tents, then the remaining 42 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be
; however, the total number of people who slept in tents was 202, not 204.
A petting zoo sells two types of tickets. The standard ticket, for admission only, costs $5. The premium ticket, which includes admission and food to give to the animals, costs $12. One Saturday, the petting zoo sold a total of 250 tickets and collected a total of $2,300 from ticket sales. Which of the following systems of equations can be used to find the number of standard tickets, s, and premium tickets, p, sold on that Saturday?
Choice A is correct. It’s given that the petting zoo sells two types of tickets, standard and premium, and that s represents the number of standard tickets sold and p represents the number of premium tickets sold. It’s also given that the petting zoo sold 250 tickets on one Saturday; thus, . It’s also given that each standard ticket costs $5 and each premium ticket costs $12. Thus, the amount collected in ticket sales can be represented by
for standard tickets and
for premium tickets. On that Saturday the petting zoo collected a total of $2,300 from ticket sales; thus,
. These two equations are correctly represented in choice A.
Choice B is incorrect. The second equation in the system represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12. Choices C and D are incorrect. The equations represent the total collected from standard and premium ticket sales as $250, not $2,300, and the total number of standard and premium tickets sold as $2,300, not $250. Additionally, the first equation in choice D represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12.
In North America, the standard width of a parking space is at least 7.5 feet and no more than 9.0 feet. A restaurant owner recently resurfaced the restaurant’s parking lot and wants to determine the number of parking spaces, n, in the parking lot that could be placed perpendicular to a curb that is 135 feet long, based on the standard width of a parking space. Which of the following describes all the possible values of n ?
Choice D is correct. Placing the parking spaces with the minimum width of 7.5 feet gives the maximum possible number of parking spaces. Thus, the maximum number that can be placed perpendicular to a 135-foot-long curb is . Placing the parking spaces with the maximum width of 9 feet gives the minimum number of parking spaces. Thus, the minimum number that can be placed perpendicular to a 135-foot-long curb is
. Therefore, if n is the number of parking spaces in the lot, the range of possible values for n is
.
Choices A and C are incorrect. These choices equate the length of the curb with the maximum possible number of parking spaces. Choice B is incorrect. This is the range of possible values for the width of a parking space instead of the range of possible values for the number of parking spaces.
Two customers purchased the same kind of bread and eggs at a store. The first customer paid dollars for loaf of bread and dozen eggs. The second customer paid dollars for loaves of bread and dozen eggs. What is the cost, in dollars, of dozen eggs?
Choice D is correct. Let represent the cost, in dollars, of loaf of bread, and let represent the cost, in dollars, of dozen eggs. It’s given that the first customer paid dollars for loaf of bread and dozen eggs. Therefore, the first customer’s purchase can be represented by the equation . It’s also given that the second customer paid dollars for loaves of bread and dozen eggs. Therefore, the second customer’s purchase can be represented by the equation . The equations and form a system of linear equations, which can be solved by elimination to find the value of . Multiplying the first equation in the system by yields . Adding to the second equation, , yields , which is equivalent to . Dividing both sides of this equation by yields . Therefore, the cost, in dollars, of dozen eggs is .
Choice A is incorrect. This is the cost, in dollars, of loaf of bread.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The equation above relates the number of minutes, x, Maria spends running each day and the number of minutes, y, she spends biking each day. In the equation, what does the number 75 represent?
The number of minutes spent running each day
The number of minutes spent biking each day
The total number of minutes spent running and biking each day
The number of minutes spent biking for each minute spent running
Choice C is correct. Maria spends x minutes running each day and y minutes biking each day. Therefore, represents the total number of minutes Maria spent running and biking each day. Because
, it follows that 75 is the total number of minutes that Maria spent running and biking each day.
Choices A and B are incorrect. The number of minutes Maria spent running each day is represented by x and need not be 75. Similarly, the number of minutes that Maria spends biking each day is represented by y and need not be 75. The number of minutes Maria spends running each day and biking each day may vary; however, the total number of minutes she spends each day on these activities is constant and equal to 75. Choice D is incorrect. The number of minutes Maria spent biking for each minute spent running cannot be determined from the information provided.
In the given equation, is a constant. The equation has infinitely many solutions. What is the value of ?
The correct answer is . It's given that the equation has infinitely many solutions. If an equation in one variable has infinitely many solutions, then the equation is true for any value of the variable. Subtracting from both sides of the given equation yields . Since this equation must be true for any value of , the value of is .
For the given linear function , which table shows three values of and their corresponding values of ?
Choice C is correct. Each of the tables shows the same three values of : , , and . Substituting for in the given function yields , or . Therefore, when , the corresponding value of is . Substituting for in the given function yields , or . Therefore, when , the corresponding value of is . Substituting for in the given function yields , or . Therefore, when , the corresponding value of is . The table in choice C shows , , and as the corresponding value of for x-values of , , and , respectively. Therefore, the table in choice C shows three values of and their corresponding values of .
Choice A is incorrect. This table shows three values of and their corresponding values of for the linear function .
Choice B is incorrect. This table shows three values of and their corresponding values of for the linear function .
Choice D is incorrect. This table shows three values of and their corresponding values of for the linear function .
If x is the solution to the equation above, what is the value of ?
Choice B is correct. Because 2 is a factor of both and 6, the expression
can be rewritten as
. Substituting
for
on the left-hand side of the given equation yields
, or
. Subtracting
from both sides of this equation yields
. Adding 11 to both sides of this equation yields
. Dividing both sides of this equation by 2 yields
.
Alternate approach: Distributing 3 to the quantity on the left-hand side of the given equation and distributing 4 to the quantity
on the right-hand side yields
, or
. Subtracting
from both sides of this equation yields
. Adding 29 to both sides of this equation yields
. Dividing both sides of this equation by 2 yields
. Therefore, the value of
is
, or
.
Choice A is incorrect. This is the value of x, not . Choices C and D are incorrect. If the value of
is
or
, it follows that the value of x is
or
, respectively. However, solving the given equation for x yields
. Therefore, the value of
can’t be
or
.
The solution to the given system of equations is . What is the value of ?
Choice A is correct. According to the first equation in the given system, . Substituting for in the second equation in the given system yields , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of , not .
Choice D is incorrect and may result from conceptual or calculation errors.
A factory makes -inch, -inch, and -inch concrete screws. During a certain day, the number of -inch concrete screws that the factory makes is times the number of -inch concrete screws, and the number of -inch concrete screws is . During this day, the factory makes concrete screws total. Which equation represents this situation?
Choice D is correct. It's given that during a certain day at a factory, the number of -inch concrete screws the factory makes is and the number of -inch concrete screws the factory makes is . It's also given that during this day the number of -inch concrete screws the factory makes is times the number of -inch concrete screws, or . Therefore, the total number of -inch, -inch, and -inch concrete screws is , or . It's given that during this day, the factory makes concrete screws total. Thus, the equation represents this situation.
Choice A is incorrect. This equation represents a situation where the total length, in inches, of all the concrete screws, rather than the total number of concrete screws, is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This equation represents a situation where the total number of -inch concrete screws and -inch concrete screws, not including the -inch concrete screws, is .
The graph of a system of linear equations is shown. What is the solution to the system?
Choice B is correct. If a point lies on both lines in the graph of a system of two linear equations, the ordered pair is a solution to the system. The graph shown is the graph of a system of two linear equations, where the two lines in the graph intersect at the point . Therefore, the point lies on both lines, so the ordered pair is the solution to the system.
Choice A is incorrect. The point lies on one, not both, of the lines in the graph shown.
Choice C is incorrect. The point lies on one, not both, of the lines in the graph shown.
Choice D is incorrect. The point lies on one, not both, of the lines in the graph shown.
Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?
Choice D is correct. Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression gives the cost, in dollars per mile, to drive the car. Multiplying
by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting
m equal to 5 gives the number of miles, m, by which he must reduce his driving.
Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5.
A certain township consists of a -hectare industrial park and a -hectare neighborhood. The total number of trees in the township is . The equation represents this situation. Which of the following is the best interpretation of in this context?
The average number of trees per hectare in the industrial park
The average number of trees per hectare in the neighborhood
The total number of trees in the industrial park
The total number of trees in the neighborhood
Choice A is correct. It's given that a certain township consists of a -hectare industrial park and a -hectare neighborhood and that the total number of trees in the township is . It's also given that the equation represents this situation. Since the total number of trees for a given area can be determined by taking the size of the area, in hectares, times the average number of trees per hectare, the best interpretation of is the number of trees in the industrial park and the best interpretation of is the number of trees in the neighborhood. Since is the size of the industrial park, in hectares, the best interpretation of is the average number of trees per hectare in the industrial park.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The shaded region shown represents the solutions to an inequality. Which ordered pair is a solution to this inequality?
Choice D is correct. Since the shaded region shown represents the solutions to an inequality, an ordered pair is a solution to the inequality if it's represented by a point in the shaded region. Of the given choices, only is represented by a point in the shaded region. Therefore, the ordered pair is a solution to this inequality.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation?
Choice A is correct. The number of containers in a shipment must have a weight less than or equal to 300 pounds. The total weight, in pounds, of detergent and fabric softener that the supplier delivers can be expressed as the weight of each container multiplied by the number of each type of container, which is 7.35d for detergent and 6.2s for fabric softener. Since this total cannot exceed 300 pounds, it follows that . Also, since the laundry service wants to buy at least twice as many containers of detergent as containers of fabric softener, the number of containers of detergent should be greater than or equal to two times the number of containers of fabric softener. This can be expressed by the inequality
.
Choice B is incorrect because it misrepresents the relationship between the numbers of each container that the laundry service wants to buy. Choice C is incorrect because the first inequality of the system incorrectly doubles the weight per container of detergent. The weight of each container of detergent is 7.35, not 14.7 pounds. Choice D is incorrect because it doubles the weight per container of detergent and transposes the relationship between the numbers of containers.
The function g is defined by . What is the value of
?
0
4
8
Choice D is correct. The value of is found by substituting 0 for x in the function g. This yields
, which can be rewritten as
.
Choice A is incorrect and may result from misinterpreting the equation as instead of
. Choice B is incorrect. This is the value of x, not
. Choice C is incorrect and may result from calculation errors.
A chemist combines water and acetic acid to make a mixture with a volume of . The volume of acetic acid in the mixture is . What is the volume of water, in , in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and acetic acid before they were mixed.)
The correct answer is . It's given that a chemist combines water and acetic acid to make a mixture with a volume of milliliters (mL) and that the volume of acetic acid in the mixture is mL. Let represent the volume of water, in mL, in the mixture. The equation represents this situation. Subtracting from both sides of this equation yields . Therefore, the volume of water, in mL, in the mixture is .
Which of the following ordered pairs is a solution to the system of inequalities above?
Choice D is correct. The solutions to the given system of inequalities is the set of all ordered pairs that satisfy both inequalities in the system. For an ordered pair to satisfy the inequality
, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the ordered pair’s x-coordinate. This is true of the ordered pair
, because
. To satisfy the inequality
, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the additive inverse of the ordered pair’s x-coordinate. This is also true of the ordered pair
. Because 0 is its own additive inverse,
is the same as
. Therefore, the ordered pair
is a solution to the given system of inequalities.
Choice A is incorrect. This ordered pair satisfies only the inequality in the given system, not both inequalities. Choice B incorrect. This ordered pair satisfies only the inequality
in the system, but not both inequalities. Choice C is incorrect. This ordered pair satisfies neither inequality.
Line in the xy-plane has a slope of and passes through the point . Which equation defines line ?
Choice D is correct. The equation that defines line in the xy-plane can be written in slope-intercept form , where is the slope of line and is its y-intercept. It’s given that line has a slope of . Therefore, . Substituting for in the equation yields , or . It’s also given that line passes through the point . Substituting for and for in the equation yields , or . Adding to both sides of this equation yields . Substituting for in the equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This equation defines a line that has a slope of , not , and passes through the point , not .
Choice C is incorrect. This equation defines a line that passes through the point , not .
A tree had a height of 6 feet when it was planted. The equation above can be used to find how many years n it took the tree to reach a height of 14 feet. Which of the following is the best interpretation of the number 2 in this context?
The number of years it took the tree to double its height
The average number of feet that the tree grew per year
The height, in feet, of the tree when the tree was 1 year old
The average number of years it takes similar trees to grow 14 feet
Choice B is correct. The height of the tree at a given time is equal to its height when it was planted plus the number of feet that the tree grew. In the given equation, 14 represents the height of the tree at the given time, and 6 represents the height of the tree when it was planted. It follows that represents the number of feet the tree grew from the time it was planted until the time it reached a height of 14 feet. Since n represents the number of years between the given time and the time the tree was planted, 2 must represent the average number of feet the tree grew each year.
Choice A is incorrect and may result from interpreting the coefficient 2 as doubling instead of as increasing by 2 each year. Choice C is incorrect. The height of the tree when it was 1 year old was feet, not 2 feet. Choice D is incorrect. No information is given to connect the growth of one particular tree to the growth of similar trees.
In the given equation, is a constant. If the equation has infinitely many solutions, what is the value of ?
The correct answer is . An equation with one variable, , has infinitely many solutions only when both sides of the equation are equal for any defined value of . It's given that , where is a constant. This equation can be rewritten as . If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of . Both sides of this equation are equal for any defined value of when . Therefore, if the equation has infinitely many solutions, the value of is .
Alternate approach: If the given equation, , has infinitely many solutions, then both sides of this equation are equal for any value of . If , then substituting for in yields , or . Dividing both sides of this equation by yields .
In the xy-plane, line passes through the points and . Which equation defines line ?
Choice D is correct. An equation defining a line in the xy-plane can be written in the form , where represents the slope and represents the y-intercept of the line. It’s given that line passes through the point ; therefore, . The slope, , of a line can be found using any two points on the line, and , and the slope formula . Substituting and for and , respectively, in the slope formula yields , or . Substituting for and for in the equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
The correct answer is . Since the number can also be written as , the given equation can also be written as . This equation is equivalent to . Therefore, the value of is . Note that 1/5 and .2 are examples of ways to enter a correct answer.
Alternate approach: Multiplying both sides of the equation by yields . Substituting for into the expression yields , or .
A machine makes large boxes or small boxes, one at a time, for a total of minutes each day. It takes the machine minutes to make a large box or minutes to make a small box. Which equation represents the possible number of large boxes, , and small boxes, , the machine can make each day?
Choice B is correct. It’s given that it takes the machine minutes to make a large box. It's also given that represents the possible number of large boxes the machine can make each day. Multiplying by gives , which represents the amount of time spent making large boxes. It’s given that it takes the machine minutes to make a small box. It's also given that represents the possible number of small boxes the machine can make each day. Multiplying by gives , which represents the amount of time spent making small boxes. Combining the amount of time spent making large boxes and small boxes yields . It’s given that the machine makes boxes for a total of minutes each day. Therefore represents the possible number of large boxes, , and small boxes, , the machine can make each day.
Choice A is incorrect and may result from associating the time of minutes with small, rather than large, boxes and the time of minutes with large, rather than small, boxes.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The perimeter of an isosceles triangle is inches. Each of the two congruent sides of the triangle has a length of inches. What is the length, in inches, of the third side?
The correct answer is . It’s given that the perimeter of an isosceles triangle is inches and that each of the two congruent sides has a length of inches. The perimeter of a triangle is the sum of the lengths of its three sides. The equation can be used to represent this situation, where is the length, in inches, of the third side. Combining like terms on the left-hand side of this equation yields . Subtracting from both sides of this equation yields . Therefore, the length, in inches, of the third side is .
If , what is the value of ?
Choice B is correct. Subtracting from each side of the given equation yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
In the given equation, b is a constant. If the equation has no solution, what is the value of b ?
2
4
6
10
Choice A is correct. This equation has no solution when there is no value of x that produces a true statement. Solving the given equation for x by dividing both sides by gives
. When
, the right-hand side of this equation will be undefined, and the equation will have no solution. Therefore, when
, there is no value of x that satisfies the given equation.
Choices B, C, and D are incorrect. Substituting 4, 6, and 10 for b in the given equation yields exactly one solution, rather than no solution, for x. For example, substituting 4 for b in the given equation yields , or
. Dividing both sides of
by 2 yields
. Similarly, if
or
,
and
, respectively.
Line in the xy-plane has a slope of and passes through the point . Which equation defines line ?
Choice D is correct. A line in the xy-plane with a slope of and a y-intercept of can be defined by an equation in the form . It’s given that line has a slope of and passes through the point . It follows that and . Substituting for and for in the equation yields . Therefore, the equation defines line .
Choice A is incorrect. This equation defines a line that has a slope of , not , and passes through the point , not .
Choice B is incorrect. This equation defines a line that has a slope of , not , and passes through the point , not .
Choice C is incorrect. This equation defines a line that passes through the point , not .
A total of paper straws of equal length were used to construct two types of polygons: triangles and rectangles. The triangles and rectangles were constructed so that no two polygons had a common side. The equation represents this situation, where is the number of triangles constructed and is the number of rectangles constructed. What is the best interpretation of in this context?
If triangles were constructed, then rectangles were constructed.
If triangles were constructed, then paper straws were used.
If triangles were constructed, then rectangles were constructed.
If triangles were constructed, then paper straws were used.
Choice A is correct. It's given that paper straws of equal length were used to construct triangles and rectangles, where no two polygons had a common side. It's also given that the equation represents this situation, where is the number of triangles constructed and is the number of rectangles constructed. The equation means that if , then . Substituting for and for in yields , or , which is true. Therefore, in this context, the equation means that if triangles were constructed, then rectangles were constructed.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
If the graph of is shifted down units in the xy-plane, what is the y-intercept of the resulting graph?
Choice A is correct. When the graph of an equation in the form , where , , and are constants, is shifted down units in the xy-plane, the resulting graph can be represented by the equation . It's given that the graph of is shifted down units in the xy-plane. Therefore, the resulting graph can be represented by the equation , or . Subtracting from both sides of this equation yields . The y-intercept of the graph of an equation in the xy-plane is the point where the line intersects the y-axis, represented by the point . Substituting for in the equation yields , or . Dividing both sides of this equation by yields . Therefore, if the graph of is shifted down units, the y-intercept of the resulting graph is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the y-intercept of the graph of shifted up, not down, units.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . For what value of does ?
The correct answer is . Substituting for in the given equation yields . Dividing the left- and right-hand sides of this equation by yields . Therefore, the value of is when .
On a car trip, Rhett and Jessica each drove for part of the trip, and the total distance they drove was under miles. Rhett drove at an average speed of , and Jessica drove at an average speed of . Which of the following inequalities represents this situation, where is the number of hours Rhett drove and is the number of hours Jessica drove?
Choice B is correct. It’s given that Rhett drove at an average speed of miles per hour and that he drove for hours. Multiplying miles per hour by hours yields miles, or the distance that Rhett drove. It’s also given that Jessica drove at an average speed of miles per hour and that she drove for hours. Multiplying miles per hour by hours yields miles, or the distance that Jessica drove. The total distance, in miles, that Rhett and Jessica drove can be represented by the expression . It’s given that the total distance they drove was under miles. Therefore, the inequality represents this situation.
Choice A is incorrect. This inequality represents a situation in which the total distance Rhett and Jessica drove was over, rather than under, miles.
Choice C is incorrect. This inequality represents a situation in which Rhett drove at an average speed of , rather than , miles per hour, Jessica drove at an average speed of , rather than , miles per hour, and the total distance they drove was over, rather than under, miles.
Choice D is incorrect. This inequality represents a situation in which Rhett drove at an average speed of , rather than , miles per hour, and Jessica drove at an average speed of , rather than , miles per hour.
A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length. Tickets for individual trips cost $1.50, $2.50, or $3.50, depending on the length of the trip. What is the minimum number of trips per month for which a monthly pass could cost less than purchasing individual tickets for trips?
The correct answer is 28. The minimum number of individual trips for which the cost of the monthly pass is less than the cost of individual tickets can be found by assuming the maximum cost of the individual tickets, $3.50. If n tickets costing $3.50 each are purchased in one month, the inequality 95 < 3.50n represents this situation. Dividing both sides of the inequality by 3.50 yields 27.14 < n, which is equivalent to n > 27.14. Since only a whole number of tickets can be purchased, it follows that 28 is the minimum number of trips.
The equation models the relationship between the number of different pieces of music a certain pianist practices, y, during an x-minute practice session. How many pieces did the pianist practice if the session lasted 30 minutes?
1
3
10
30
Choice B is correct. It’s given that the equation models the relationship between the number of different pieces of music a certain pianist practices, y, and the number of minutes in a practice session, x. Since it’s given that the session lasted 30 minutes, the number of pieces the pianist practiced can be found by substituting 30 for x in the given equation, which yields
, or
.
Choices A and C are incorrect and may result from misinterpreting the values in the equation. Choice D is incorrect. This is the given value of x, not the value of y.
A bakery sells trays of cookies. Each tray contains at least 50 cookies but no more than 60. Which of the following could be the total number of cookies on 4 trays of cookies?
165
205
245
285
Choice B is correct. If each tray contains the least number of cookies possible, 50 cookies, then the least number of cookies possible on 4 trays is 50 × 4 = 200 cookies. If each tray contains the greatest number of cookies possible, 60 cookies, then the greatest number of cookies possible on 4 trays is 60 × 4 = 240 cookies. If the least number of cookies on 4 trays is 200 and the greatest number of cookies is 240, then 205 could be the total number of cookies on these 4 trays of cookies because .
Choices A, C, and D are incorrect. The least number of cookies on 4 trays is 200 cookies, and the greatest number of cookies on 4 trays is 240 cookies. The choices 165, 245, and 285 are each either less than 200 or greater than 240; therefore, they cannot represent the total number of cookies on 4 trays.
How many solutions does the equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice C is correct. Applying the distributive property to each side of the given equation yields . Applying the commutative property of addition to the right-hand side of this equation yields . Since the two sides of the equation are equivalent, this equation is true for any value of . Therefore, the given equation has infinitely many solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of x?
The correct answer is . It’s given in the first equation of the system that . Substituting for in the second equation of the system yields . Combining like terms on the left-hand side of this equation yields . Therefore, the value of is .
Hydrogen is placed inside a container and kept at a constant pressure. The graph shows the estimated volume , in liters, of the hydrogen when its temperature is kelvins.
What is the estimated volume, in liters, of the hydrogen when its temperature is kelvins?
Choice C is correct. For the graph shown, the x-axis represents temperature, in kelvins, and the y-axis represents volume, in liters. Therefore, the estimated volume, in liters, of the hydrogen when its temperature is kelvins is represented by the y-coordinate of the point on the graph that has an x-coordinate of . The point on the graph with an x-coordinate of has a y-coordinate of . Therefore, the estimated volume, in liters, of the hydrogen when its temperature is kelvins is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
What is the solution to the given system of equations?
Choice A is correct. The second equation in the given system defines the value of as . Substituting for into the first equation yields or . Dividing each side of this equation by yields . Substituting for in the second equation yields or . Therefore, the solution to the given system of equations is .
Choice B is incorrect. Substituting for and for in the second equation yields , which is not true. Therefore, is not a solution to the given system of equations.
Choice C is incorrect. Substituting for and for in the second equation yields , which is not true. Therefore, is not a solution to the given system of equations.
Choice D is incorrect. Substituting for and for in the second equation yields , which is not true. Therefore, is not a solution to the given system of equations.
The line graphed in the xy-plane below models the total cost, in dollars, for a cab ride, y, in a certain city during nonpeak hours based on the number of miles traveled, x.
According to the graph, what is the cost for each additional mile traveled, in dollars, of a cab ride?
$2.00
$2.60
$3.00
$5.00
Choice A is correct. The cost of each additional mile traveled is represented by the slope of the given line. The slope of the line can be calculated by identifying two points on the line and then calculating the ratio of the change in y to the change in x between the two points. Using the points and
, the slope is equal to
, or 2. Therefore, the cost for each additional mile traveled of the cab ride is $2.00.
Choice B is incorrect and may result from calculating the slope of the line that passes through the points and
. However,
does not lie on the line shown. Choice C is incorrect. This is the y-coordinate of the y-intercept of the graph and represents the flat fee for a cab ride before the charge for any miles traveled is added. Choice D is incorrect. This value represents the total cost of a 1-mile cab ride.
In the xy-plane, line k intersects the y-axis at the point and passes through the point
. If the point
lies on line k, what is the value of w ?
The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point
, it follows that
is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form
, where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of line k can be written as
, or
. The value of m can be found by substituting the x- and y-coordinates from a point on the line, such as
, for x and y, respectively. This results in
. Solving this equation for m gives
. Therefore, an equation of line k is
. The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives
, or
. Since w is the y-coordinate of this point,
.
Which point is a solution to the given system of inequalities in the xy-plane?
Choice D is correct. A point is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for and , respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, and , for and , respectively, in the first inequality in the given system, , yields , or , which is true. Substituting for and for in the second inequality in the given system, , yields , or , which is true. Therefore, the point is a solution to the given system of inequalities in the xy-plane.
Choice A is incorrect. Substituting for and for in the inequality yields , or , which is not true.
Choice B is incorrect. Substituting for and for in the inequality yields , or , which is not true.
Choice C is incorrect. Substituting for and for in the inequality yields , or , which is not true.
A team of workers has been moving cargo off of a ship. The equation below models the approximate number of tons of cargo, y, that remains to be moved x hours after the team started working.
The graph of this equation in the xy-plane is a line. What is the best interpretation of the x-intercept in this context?
The team will have moved all the cargo in about 4.8 hours.
The team has been moving about 4.8 tons of cargo per hour.
The team has been moving about 25 tons of cargo per hour.
The team started with 120 tons of cargo to move.
Choice A is correct. The x-intercept of the line with equation y = 120 – 25x can be found by substituting 0 for y and finding the value of x. When y = 0, x = 4.8, so the x-intercept is at (4.8, 0). Since y represents the number of tons of cargo remaining to be moved x hours after the team started working, it follows that the x-intercept refers to the team having no cargo remaining to be moved after 4.8 hours. In other words, the team will have moved all of the cargo after about 4.8 hours.
Choice B is incorrect and may result from incorrectly interpreting the value 4.8. Choices C and D are incorrect and may result from misunderstanding the x-intercept. These statements are accurate but not directly relevant to the x-intercept.
At how many points do the graphs of the given equations intersect in the xy-plane?
Zero
Exactly one
Exactly two
Infinitely many
Choice A is correct. A system of two linear equations in two variables, and , has zero points of intersection if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, , are distinct if the y-coordinates of their y-intercepts, , are different and are parallel if their slopes, , are the same. For the two equations in the given system, and , the values of are and , respectively, and the values of are both . Since the values of are different, the graphs of these lines have different y-coordinates of the y-intercept and are distinct. Since the values of are the same, the graphs of these lines have the same slope and are parallel. Therefore, the graphs of the given equations are lines that intersect at zero points in the xy-plane.
Choice B is incorrect. The graphs of a system of two linear equations have exactly one point of intersection if the lines represented by the equations have different slopes. Since the given equations represent lines with the same slope, there is not exactly one intersection point.
Choice C is incorrect. The graphs of a system of two linear equations can never have exactly two intersection points.
Choice D is incorrect. The graphs of a system of two linear equations have infinitely many intersection points when the lines represented by the equations have the same slope and the same y-coordinate of the y-intercept. Since the given equations represent lines with different y-coordinates of their y-intercepts, there are not infinitely many intersection points.
Which of the following is an equation of the graph shown in the xy-plane above?
Choice A is correct. The slope of the line can be found by choosing any two points on the line, such as (4, –2) and (0, –1). Subtracting the y-values results in –2 – (–1) = –1, the change in y. Subtracting the x-values results in 4 – 0 = 4, the change in x. Dividing the change in y by the change in x yields , the slope. The line intersects the y-axis at (0, –1), so –1 is the y-coordinate of the y-intercept. This information can be expressed in slope-intercept form as the equation
.
Choice B is incorrect and may result from incorrectly calculating the slope and then misidentifying the slope as the y-intercept. Choice C is incorrect and may result from misidentifying the slope as the y-intercept. Choice D is incorrect and may result from incorrectly calculating the slope.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by yields , or . Adding this equation to the first equation in the given system, , yields , or . Subtracting from both sides of this equation yields , or . If the given system has no solution, then the equation has no solution. If this equation has no solution, the coefficients of on each side of the equation, and , must be equal, which yields the equation . Dividing both sides of this equation by yields . Thus, if the system has no solution, the value of is .
Alternate approach: A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in the form , where , , and are constant terms, are parallel if the ratio of the x-coefficients is equal to the ratio of the y-coefficients, and distinct if the ratio of the x-coefficients are not equal to the ratio of the constant terms. Subtracting from both sides of the first equation in the given system yields , or . Subtracting from both sides of the second equation in the given system yields , or . The ratio of the x-coefficients for these equations is , or . The ratio of the y-coefficients for these equations is . The ratio of the constant terms for these equations is , or . Since the ratio of the x-coefficients, , is not equal to the ratio of the constants, , the lines represented by the equations are distinct. Setting the ratio of the x-coefficients equal to the ratio of the y-coefficients yields . Multiplying both sides of this equation by yields , or . Therefore, when , the lines represented by these equations are parallel. Thus, if the system has no solution, the value of is .
Which table gives three values of and their corresponding values of for the given equation?
Choice D is correct. Each of the tables gives the same three values of : , , and . Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, when , the corresponding value of for the given equation is . The table in choice D gives x-values of , , and and corresponding y-values of , , and , respectively. Therefore, the table in choice D gives three values of and their corresponding values of for the given equation.
Choice A is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice B is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice C is incorrect. This table gives three values of and their corresponding values of for the equation .
A contract for a certain service requires a onetime activation cost of and a monthly cost of . Which equation represents this situation, where is the total cost, in dollars, of this service contract for months?
Choice C is correct. It's given that this service contract requires a monthly cost of . A monthly cost of for months results in a cost of . It's also given that this service contract requires a onetime activation cost of . Adding the onetime activation cost to the monthly cost of the service contract for months yields the total cost , in dollars, of this service contract for months. Therefore, this situation can be represented by the equation .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
An employee at a restaurant prepares sandwiches and salads. It takes the employee minutes to prepare a sandwich and minutes to prepare a salad. The employee spends a total of minutes preparing sandwiches and salads. Which equation represents this situation?
Choice B is correct. It’s given that the employee takes minutes to prepare a sandwich. Multiplying by the number of sandwiches, , yields , the amount of time the employee spends preparing sandwiches. It’s also given that the employee takes minutes to prepare a salad. Multiplying by the number of salads, , yields , the amount of time the employee spends preparing salads. It follows that the total amount of time, in minutes, the employee spends preparing sandwiches and salads is . It's given that the employee spends a total of minutes preparing sandwiches and salads. Thus, the equation represents this situation.
Choice A is incorrect. This equation represents a situation where it takes the employee minutes, rather than minutes, to prepare a sandwich and minutes, rather than minutes, to prepare a salad.
Choice C is incorrect. This equation represents a situation where it takes the employee minute, rather than minutes, to prepare a sandwich and minute, rather than minutes, to prepare a salad.
Choice D is incorrect. This equation represents a situation where it takes the employee minutes, rather than minutes, to prepare a sandwich and minutes, rather than minutes, to prepare a salad.
The function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. If a temperature increased by kelvins, by how much did the temperature increase, in degrees Fahrenheit?
Choice A is correct. It’s given that the function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. A temperature that increased by kelvins means that the value of increased by kelvins. It follows that an increase in by increases by , or . Therefore, if a temperature increased by kelvins, the temperature increased by degrees Fahrenheit.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the equation above, what is the value of x ?
25
24
16
15
Choice A is correct. Multiplying both sides of the equation by 5 results in . Dividing both sides of the resulting equation by 4 results in
.
Choice B is incorrect and may result from adding 20 and 4. Choice C is incorrect and may result from dividing 20 by 5 and then multiplying the result by 4. Choice D is incorrect and may result from subtracting 5 from 20.
The pressure exerted on a scuba diver at sea level is . For each foot the scuba diver descends below sea level, the pressure exerted on the scuba diver increases by . What is the total pressure, in , exerted on the scuba diver at feet below sea level?
Choice A is correct. It's given that the pressure exerted on a scuba diver at sea level is . It's also given that for each foot the scuba diver descends below sea level, the pressure exerted on the scuba diver increases by . The total pressure, in , exerted on the scuba diver at feet below sea level can be represented by the expression . Substituting for in this expression yields , or . Therefore, the total pressure exerted on the scuba diver at feet below sea level is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the pressure, in , exerted on the scuba diver at sea level, not at feet below sea level.
Choice D is incorrect. This is the rate by which the pressure, in , exerted on the scuba diver increases for each foot the scuba diver descends below sea level.
A truck can haul a maximum weight of pounds. During one trip, the truck will be used to haul a -pound piece of equipment as well as several crates. Some of these crates weigh pounds each and the others weigh pounds each. Which inequality represents the possible combinations of the number of -pound crates, , and the number of -pound crates, , the truck can haul during one trip if only the piece of equipment and the crates are being hauled?
Choice A is correct. It's given that a truck can haul a maximum of pounds. It's also given that during one trip, the truck will be used to haul a -pound piece of equipment as well as several crates. It follows that the truck can haul at most , or , pounds of crates. Since represents the number of -pound crates, the expression represents the weight of the -pound crates. Since represents the number of -pound crates, represents the weight of the -pound crates. Therefore, represents the total weight of the crates the truck can haul. Since the truck can haul at most pounds of crates, the total weight of the crates must be less than or equal to pounds, or .
Choice B is incorrect. This represents the possible combinations of the number of -pound crates, , and the number of -pound crates, , the truck can haul during one trip if it can haul a minimum, not a maximum, of pounds.
Choice C is incorrect. This represents the possible combinations of the number of -pound crates, , and the number of -pound crates, , the truck can haul during one trip if only crates are being hauled.
Choice D is incorrect. This represents the possible combinations of the number of -pound crates, , and the number of -pound crates, , the truck can haul during one trip if it can haul a minimum, not a maximum, weight of pounds and only crates are being hauled.
Which of the following graphs in the xy-plane could be used to solve the system of equations above?
Choice C is correct. The graph of a system of equations is the graph that shows the lines represented by each of the equations in the system. The x-intercept of the graph of each given equation can be found by substituting 0 for y in each equation: , or
, and
, or
. The y-intercept of the graph of each equation can be found by substituting 0 for x in each equation:
, or
, and
or
. Using these x- and y- intercept values, the line that has equation
passes through the points
and
, and the line that has equation
passes through the points
and
. Only the lines in choice C pass through these points and can be used to solve the given system of equations.
Choices A, B, and D are incorrect. In choices A and B, neither line passes through and
or
and
. In choice D, although one line passes through
and
the other line doesn’t pass through
and
.
In the given equation, is a constant. The equation has no solution. What is the value of ?
The correct answer is . It's given that the equation has no solution. A linear equation in the form , where , , , and are constants, has no solution only when the coefficients of on each side of the equation are equal and the constant terms aren't equal. Dividing both sides of the given equation by yields , or . Since the coefficients of on each side of the equation must be equal, it follows that the value of is . Note that 16/17, .9411, .9412, and 0.941 are examples of ways to enter a correct answer.
The function is defined by . What is the x-intercept of the graph of in the xy-plane?
Choice D is correct. The given function is a linear function. Therefore, the graph of in the xy-plane has one x-intercept at the point , where is a constant. Substituting for and for in the given function yields . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the x-intercept of the graph of in the xy-plane is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The table shows four values of and their corresponding values of . There is a linear relationship between and that is defined by the equation , where is a constant. What is the value of ?
The correct answer is . It's given that is defined by the equation , where is a constant. It's also given in the table that when , . Substituting for and for in the equation yields, . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the value of is .
Julissa needs at least hours of flight time to get her private pilot certification. If Julissa already has hours of flight time, what is the minimum number of additional hours of flight time Julissa needs to get her private pilot certification?
Choice A is correct. It's given that Julissa already has hours of flight time. Let represent the number of additional hours of flight time Julissa needs to get her private pilot certification. After completing hours of flight time, Julissa will have completed a total of hours of flight time. It's given that Julissa needs at least hours of flight time to get her private pilot certification. Therefore, . Subtracting from both sides of this inequality yields . Thus, is the minimum number of additional hours of flight time Julissa needs to get her private pilot certification.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the number of hours of flight time Julissa already has, rather than the minimum number of additional hours of flight time Julissa needs.
Choice D is incorrect. This is the number of hours of flight time Julissa will have if she completes more hours of flight time, rather than the minimum number of additional hours of flight time Julissa needs.
The graph of the linear function is shown. If and are positive constants, which equation could define ?
Choice A is correct. It’s given that the graph of the linear function is shown. This means that the graph of can be translated down units to create the graph of and the y-coordinate of every point on the graph of can be decreased by to find the resulting point on the graph of . The y-intercept of the graph of is . Translating the graph of down units results in a y-intercept of the graph of at the point , or . The graph of slants down from left to right, so the slope of the graph is negative. The translation of a linear graph changes its position, but does not change its slope. It follows that the slope of the graph of is also negative. The equation of a linear function can be written in the form , where is the y-coordinate of the y-intercept and is the slope of the graph of . It's given that and are positive constants. Since the y-coordinate of the y-intercept and the slope of the graph of are both negative, it follows that could define .
Choice B is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept.
Choice C is incorrect. This could define a linear function where its graph has a positive, not negative, slope.
Choice D is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept and a positive, not negative, slope.
The solution to the given system of equations is . What is the value of ?
Choice D is correct. It's given by the first equation in the given system of equations that . Substituting for in the second equation in the given system yields , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not the value of .
Choice C is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. It's given by the first equation in the system that . Substituting for in the second equation in the system yields . Multiplying the left-hand side of this equation by and the right-hand side by yields . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A salesperson’s total earnings consist of a base salary of dollars per year, plus commission earnings of of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least times and at most times the base salary. Which of the following inequalities represents all possible values of total sales , in dollars, the salesperson can make this year in order to meet that goal?
Choice B is correct. It’s given that a salesperson's total earnings consist of a base salary of dollars per year plus commission earnings of of the total sales the salesperson makes during the year. If the salesperson makes dollars in total sales this year, the salesperson’s total earnings can be represented by the expression . It’s also given that the salesperson has a goal for the total earnings to be at least times and at most times the base salary, which can be represented by the expressions and , respectively. Therefore, this situation can be represented by the inequality . Subtracting from each part of this inequality yields . Dividing each part of this inequality by yields . Therefore, the inequality represents all possible values of total sales , in dollars, the salesperson can make this year in order to meet their goal.
Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least times and at most times, rather than at least times and at most times, the base salary.
Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least times and at most times the base salary.
Choice D is incorrect. This inequality represents a situation in which the total earnings are at least times and at most times, rather than at least times and at most times, the base salary.
For the linear function , the table shows three values of and their corresponding values of . If , which equation defines ?
Choice B is correct. An equation that defines a linear function can be written in the form , where and are constants. It's given in the table that when , . Substituting for and for in the equation yields , or . Adding to both sides of this equation yields . Substituting for in the equation yields . It's also given in the table that when , . Substituting for and for in the equation yields , or . Multiplying both sides of this equation by yields . Substituting for in the equation yields , or . If , substituting for in this equation yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is an equation that defines the linear function , not .
Which of the following is equivalent to ?
Choice D is correct. Dividing each side of the original equation by yields
, which simplifies to
.
Choice A is incorrect. Dividing each side of the original equation by gives
, which is not equivalent to
. Choice B is incorrect. Dividing each side of the original equation by
gives
, which is not equivalent to
. Choice C is incorrect. Dividing each side of the original equation by
gives
, which is not equivalent to
.
What is the slope of the graph of in the xy-plane?
Choice D is correct. A linear equation can be written in the form , where is the slope of the graph of the equation in the xy-plane and is the y-intercept. Subtracting from each side of the given equation, , yields . Dividing each side of this equation by yields . This equation is in the form , where . Therefore, the slope of the graph of the given equation in the xy-plane is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
In the xy-plane, line has a slope of and an x-intercept of . What is the y-coordinate of the y-intercept of line ?
The correct answer is . A line in the xy-plane can be represented by the equation , where is the slope of the line and is the y-coordinate of the y-intercept. It's given that line has a slope of . Therefore, . It's also given that line has an x-intercept of . Therefore, when , . Substituting for , for , and for in the equation yields , which is equivalent to . Subtracting from both sides of this equation yields . Therefore, the y-coordinate of the y-intercept of line is .
One pound of grapes costs $2. At this rate, how many dollars will c pounds of grapes cost?
Choice A is correct. If one pound of grapes costs $2, two pounds of grapes will cost 2 times $2, three pounds of grapes will cost 3 times $2, and so on. Therefore, c pounds of grapes will cost c times $2, which is 2c dollars.
Choice B is incorrect and may result from incorrectly adding instead of multiplying. Choice C is incorrect and may result from assuming that c pounds cost $2, and then finding the cost per pound. Choice D is incorrect and could result from incorrectly assuming that 2 pounds cost $c, and then finding the cost per pound.
The graph of the function , where , gives the total cost , in dollars, for a certain video game system and games. What is the best interpretation of the slope of the graph in this context?
Each game costs .
The video game system costs .
The video game system costs .
Each game costs .
Choice A is correct. The given graph is a line, and the slope of a line is defined as the change in the value of for each increase in the value of by . It’s given that represents the total cost, in dollars, and that represents the number of games. Therefore, the change in the value of for each increase in the value of by represents the change in total cost, in dollars, for each increase in the number of games by . In other words, the slope represents the cost, in dollars, per game. The graph shows that when the value of increases from to , the value of increases from to . It follows that the slope is , or the cost per game is . Thus, the best interpretation of the slope of the graph is that each game costs .
Choice B is incorrect. This is an interpretation of the y-intercept of the graph rather than the slope of the graph.
Choice C is incorrect. The slope of the graph is the cost per game, not the cost of the video game system.
Choice D is incorrect. Each game costs , not .
A bus traveled on the highway and on local roads to complete a trip of . The trip took . The bus traveled at an average speed of on the highway and an average speed of on local roads. If is the time, in hours, the bus traveled on the highway and is the time, in hours, it traveled on local roads, which system of equations represents this situation?
Choice B is correct. If the bus traveled at an average speed of on the highway for hours, then the bus traveled miles on the highway. If the bus traveled at an average speed of on local roads for hours, then the bus traveled miles on local roads. It's given that the trip was miles. This can be represented by the equation . It's also given that the trip took hours. This can be represented by the equation . Therefore, the system consisting of the equations and represents this situation.
Choice A is incorrect. This system of equations represents a situation where the trip was miles and took hours.
Choice C is incorrect. This system of equations represents a situation where the trip was miles and took hours, and the bus traveled at an average speed of on the highway and on local roads.
Choice D is incorrect. This system of equations represents a situation where the bus traveled at an average speed of on the highway and on local roads.
On a 210-mile trip, Cameron drove at an average speed of 60 miles per hour for the first x hours. He then completed the trip, driving at an average speed of 50 miles per hour for the remaining y hours. If , what is the value of y ?
The correct answer is 3. It’s given that Cameron drove 60 miles per hour for x hours; therefore, the distance driven at this speed can be represented by . He then drove 50 miles per hour for y hours; therefore, the distance driven at this speed can be represented by
. Since Cameron drove 210 total miles, the equation
represents this situation. If
, substitution yields
, or
. Subtracting 60 from both sides of this equation yields
. Dividing both sides of this equation by 50 yields
.
For the linear function , the graph of in the xy-plane has a slope of and has a y-intercept at . Which equation defines ?
Choice D is correct. An equation defining the linear function can be written in the form , where is the slope and is the y-intercept of the graph of in the xy-plane. It’s given that the graph of has a slope of . Therefore, . It’s also given that the graph of has a y-intercept at . Therefore, . Substituting for and for in the equation yields . Thus, the equation that defines is .
Choice A is incorrect. For this function, the graph of in the xy-plane has a slope of , not .
Choice B is incorrect. For this function, the graph of in the xy-plane has a slope of , not .
Choice C is incorrect. For this function, the graph of in the xy-plane has a slope of , not .
Ty set a goal to walk at least kilometers every day to prepare for a multiday hike. On a certain day, Ty plans to walk at an average speed of kilometers per hour. What is the minimum number of hours Ty must walk on that day to fulfill the daily goal?
Choice B is correct. It's given that Ty plans to walk at an average speed of kilometers per hour. The number of kilometers Ty will walk is determined by the expression , where is the number of hours Ty walks. The given goal of at least kilometers means that the inequality represents the situation. Dividing both sides of this inequality by gives , which corresponds to a minimum of hours Ty must walk.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the solution to the given system of equations?
Choice B is correct. Since it’s given that , substituting 1 for x in the first equation yields
. Simplifying the right-hand side of this equation yields
, or
. Therefore, the ordered pair
is a solution to the given system of equations.
Choice A is incorrect and may result from a calculation error when substituting 1 for x in the first equation. Choices C and D are incorrect. Because it’s given that , x cannot equal 2 as stated in these ordered pairs.
A system of two linear equations is graphed in the xy-plane below.
Which of the following points is the solution to the system of equations?
Choice A is correct. The solution to this system of linear equations is the point that lies on both lines graphed, or the point of intersection of the two lines. According to the graphs, the point of intersection occurs when and
, or at the point
.
Choices B and D are incorrect. Each of these points lies on one line, but not on both lines in the xy-plane. Choice C is incorrect. This point doesn’t lie on either of the lines graphed in the xy-plane.
The function gives the monthly fee , in dollars, a facility charges to keep crates in storage. What is the monthly fee, in dollars, the facility charges to keep crates in storage?
The correct answer is . It’s given that the function gives the monthly fee, in dollars, a facility charges to keep crates in storage. Substituting for in this function yields , or . Therefore, the monthly fee, in dollars, the facility charges to keep crates in storage is .
The function is defined by . What is the value of ?
Choice B is correct. It’s given that function is defined by . The value of can be found by substituting for in the given function, which yields , or , which is equivalent to . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
The functions and are defined as and . If the function is defined as , what is the x-coordinate of the x-intercept of the graph of in the xy-plane?
The correct answer is . It's given that the functions and are defined as and . If the function is defined as , then substituting for and for in this function yields . This can be rewritten as , or . The x-intercept of a graph in the xy-plane is the point on the graph where . The equation representing the graph of is . Substituting for in this equation yields . Subtracting from both sides of this equation yields , or . Therefore, the x-coordinate of the x-intercept of the graph of in the xy-plane is .
A principal used a total of flags that were either blue or yellow for field day. The principal used blue flags. How many yellow flags were used?
Choice A is correct. It's given that a principal used a total of blue flags and yellow flags. It's also given that of the flags used, flags were blue. Subtracting the number of blue flags used from the total number of flags used results in the number of yellow flags used. It follows that the number of yellow flags used is , or .
Choice B is incorrect. This is the number of blue flags used.
Choice C is incorrect. This is the total number of flags used.
Choice D is incorrect and may result from conceptual or calculation errors.
During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month?
80
100
120
160
Choice D is correct. The number of hours Morgan spent running or biking can be calculated by dividing the distance she traveled during that activity by her speed, in miles per hour, for that activity. So the number of hours she ran can be represented by the expression , and the number of hours she biked can be represented by the expression
. It’s given that she biked for twice as many hours as she ran, so this can be represented by the equation
, which can be rewritten as
. It’s also given that she ran r miles and biked b miles, and that she ran and biked a total of 200 miles. This can be represented by the equation
. Substituting
for b in this equation yields
, or
. Solving for r yields
. Determining the number of miles she biked, b, can be found by substituting 40 for r in
, which yields
. Solving for b yields
.
Choices A, B, and C are incorrect because they don’t satisfy that Morgan biked for twice as many hours as she ran. In choice A, if she biked 80 miles, then she ran 120 miles, which means she biked for 8 hours and ran for 24 hours. In choice B, if she biked 100 miles, then she ran 100 miles, which means she biked for 10 hours and ran for 20 hours. In choice C, if she biked 120 miles, then she ran for 80 miles, which means she biked for 12 hours and ran for 16 hours.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. Adding the second equation in the given system to the first equation in the given system yields . Adding like terms in this equation yields . Thus, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect and may result from conceptual or calculation errors.
What value of is the solution to the given equation?
Choice A is correct. Dividing both sides of the given equation by yields . Therefore, is the solution to the given equation.
Choice B is incorrect. This is the solution to the equation .
Choice C is incorrect. This is the solution to the equation .
Choice D is incorrect. This is the solution to the equation .
The combined original price for a mirror and a vase is . After a discount to the mirror and a discount to the vase are applied, the combined sale price for the two items is . Which system of equations gives the original price , in dollars, of the mirror and the original price , in dollars, of the vase?
Choice C is correct. It’s given that represents the original price, in dollars, of the mirror, and represents the original price, in dollars, of the vase. It's also given that the combined original price for the mirror and the vase is . This can be represented by the equation . After a discount to the mirror is applied, the sale price of the mirror is of its original price. This can be represented by the expression . After a discount to the vase is applied, the sale price of the vase is of its original price. This can be represented by the expression . It’s given that the combined sale price for the two items is . This can be represented by the equation . Therefore, the system of equations consisting of the equations and gives the original price , in dollars, of the mirror and the original price , in dollars, of the vase.
Choice A is incorrect. The second equation in this system of equations represents a discount to the mirror and a discount to the vase.
Choice B is incorrect. The second equation in this system of equations represents a discount to the mirror and a discount to the vase.
Choice D is incorrect. The second equation in this system of equations represents a discount to the mirror and a discount to the vase.
The graph shown models the number of candy bars a certain machine wraps with a label in seconds.
According to the graph, what is the estimated number of candy bars the machine wraps with a label per second?
Choice B is correct. For the graph shown, the x-axis represents time, in seconds, and the y-axis represents the number of candy bars wrapped. The slope of a line in the xy-plane is the change in for each -unit increase in . It follows that the slope of the graph shown represents the estimated number of candy bars the machine wraps with a label per second. The slope, , of a line in the xy-plane can be found using any two points, and , on the line and the slope formula . The graph shown passes through the points and . Substituting these points for and , respectively, in the slope formula yields , which is equivalent to , or . Therefore, the estimated number of candy bars the machine wraps with a label per second is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For the linear function , is a constant and . What is the value of ?
The correct answer is . It’s given that . Therefore, for the given function , when , . Substituting for and for in the given function, , yields , or . Therefore, the value of is .
If the system of equations above has solution , what is the value of
?
0
9
18
38
Choice C is correct. Adding the given equations yields 9x + 9y = 162. Dividing each side of the equation 9x + 9y = 162 by 9 gives x + y = 18.
Choice A is incorrect and may result from incorrectly adding the equations. Choice B is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This value is equivalent to y – x.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Subtracting the second equation from the first equation in the given system of equations yields , which is equivalent to, or . Dividing each side of this equation by yields .
The graph of the function is shown, where . What is the y-intercept of the graph?
Choice B is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function shown intersects the y-axis at the point . Therefore, the y-intercept of the graph is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The table shows three values of and their corresponding values of . Which equation represents the linear relationship between and ?
Choice A is correct. The linear relationship between and can be represented by the equation , where is the slope of the line in the xy-plane that represents the relationship, and is the y-coordinate of the y-intercept. The slope can be computed using any two points on the line. The slope of a line between any two points, and , on the line can be calculated using the slope formula, . In the given table, each value of and its corresponding value of can be represented by a point . In the given table, when the value of is , the corresponding value of is and when the value of is , the corresponding value of is . Therefore, the points and are on the line. Substituting and for and , respectively, in the slope formula yields , or . Substituting for in the equation yields . Substituting the first value of in the table, , and its corresponding value of , , for and , respectively, in this equation yields , or . Subtracting from both sides of this equation yields . Substituting for in the equation yields . Therefore, the equation represents the linear relationship between and .
Choice B is incorrect. For this relationship, when the value of is , the corresponding value of is , not .
Choice C is incorrect. For this relationship, when the value of is , the corresponding value of is , not .
Choice D is incorrect. For this relationship, when the value of is , the corresponding value of is , not .
A number is at most less than times the value of . If the value of is , what is the greatest possible value of ?
The correct answer is . It's given that a number is at most less than times the value of . Therefore, is less than or equal to less than times the value of . The expression represents times the value of . The expression represents less than times the value of . Therefore, is less than or equal to . This can be shown by the inequality . Substituting for in this inequality yields or, . Therefore, if the value of is , the greatest possible value of is .
Caleb used juice to make popsicles. The function approximates the volume, in fluid ounces, of juice Caleb had remaining after making popsicles. Which statement is the best interpretation of the y-intercept of the graph of in the xy-plane in this context?
Caleb used approximately fluid ounces of juice for each popsicle.
Caleb had approximately fluid ounces of juice when he began to make the popsicles.
Caleb had approximately fluid ounces of juice when he began to make the popsicles.
Caleb used approximately fluid ounces of juice for each popsicle.
Choice C is correct. An equation that defines a linear function can be written in the form , where represents the slope and represents the y-intercept, , of the line of in the xy-plane. The function is linear. Therefore, the graph of the given function in the xy-plane has a y-intercept of . It’s given that gives the approximate volume, in fluid ounces, of juice Caleb had remaining after making popsicles. It follows that the y-intercept of means that Caleb had approximately fluid ounces of juice remaining after making popsicles. In other words, Caleb had approximately fluid ounces of juice when he began to make the popsicles.
Choice A is incorrect. This is an interpretation of the slope, rather than the y-intercept, of the graph of in the xy-plane.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
At a state fair, attendees can win tokens that are worth a different number of points depending on the shape. One attendee won square tokens and circle tokens worth a total of points. The equation represents this situation. How many more points is a circle token worth than a square token?
Choice D is correct. It’s given that the equation represents this situation, where is the number of square tokens won, is the number of circle tokens won, and is the total number of points the tokens are worth. It follows that represents the total number of points the square tokens are worth. Therefore, each square token is worth points. It also follows that represents the total number of points the circle tokens are worth. Therefore, each circle token is worth points. Since a circle token is worth points and a square token is worth points, then a circle token is worth , or , more points than a square token.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the number of points a circle token is worth.
Choice C is incorrect. This is the number of points a square token is worth.
According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the United States from September 1, 2014, to December 1, 2014, is modeled by the function F defined below, where is the average price per gallon x months after September 1.
The constant 2.74 in this function estimates which of the following?
The average monthly decrease in the price per gallon
The difference in the average price per gallon from September 1, 2014, to December 1, 2014
The average price per gallon on September 1, 2014
The average price per gallon on December 1, 2014
Choice D is correct. Since 2.74 is a constant term, it represents an actual price of gas rather than a measure of change in gas price. To determine what gas price it represents, find x such that F(x) = 2.74, or 2.74 = 2.74 – 0.19(x – 3). Subtracting 2.74 from both sides gives 0 = –0.19(x – 3). Dividing both sides by –0.19 results in 0 = x – 3, or x = 3. Therefore, the average price of gas is $2.74 per gallon 3 months after September 1, 2014, which is December 1, 2014.
Choice A is incorrect. Since 2.74 is a constant, not a multiple of x, it cannot represent a rate of change in price. Choice B is incorrect. The difference in the average price from September 1, 2014, to December 1, 2014, is F(3) – F(0) = 2.74 – 0.19(3 – 3) – (2.74 – 0.19(0 – 3)) = 2.74 – (2.74 + 0.57) = –0.57, which is not 2.74. Choice C is incorrect. The average price per gallon on September 1, 2014, is F(0) = 2.74 – 0.19(0 – 3) = 2.74 + 0.57 = 3.31, which is not 2.74.
In the xy-plane, the graph of intersects the graph of
at the point
. What is the value of a ?
3
6
9
12
Choice C is correct. Since the graph of intersects the graph of
at the point
, the ordered pair
is the solution to the system of linear equations consisting of
and
, and the value of a is the value of x in the solution of this system. Since both
and
are equal to y, it follows that
. Subtracting x from and adding 6 to both sides of the equation yields
. Therefore, the value of a is 9.
Choices A and B are incorrect and may result from a calculation or conceptual error in solving the system of equations consisting of and
. Choice D is incorrect. This is the value of b, not a.
In the xy-plane, the graph of the linear function contains the points and . Which equation defines , where ?
Choice C is correct. In the xy-plane, the graph of a linear function can be written in the form , where represents the slope and represents the y-intercept of the graph of . It’s given that the graph of the linear function , where , in the xy-plane contains the point . Thus, . The slope of the graph of a line containing any two points and can be found using the slope formula, . Since it’s given that the graph of the linear function contains the points and , it follows that the slope of the graph of the line containing these points is , or . Substituting for and for in yields .
Choice A is incorrect. This function represents a graph with a slope of and a y-intercept of .
Choice B is incorrect. This function represents a graph with a slope of and a y-intercept of .
Choice D is incorrect. This function represents a graph with a slope of and a y-intercept of .
| Number of cars | Maximum number of passengers and crew |
|---|---|
The table shows the linear relationship between the number of cars, , on a commuter train and the maximum number of passengers and crew, , that the train can carry. Which equation represents the linear relationship between and ?
Choice A is correct. It's given that there is a linear relationship between the number of cars, , on a commuter train and the maximum number of passengers and crew, , that the train can carry. It follows that this relationship can be represented by an equation of the form , where is the rate of change of in this relationship and is a constant. The rate of change of in this relationship can be calculated by dividing the difference in any two values of by the difference in the corresponding values of . Using two pairs of values given in the table, the rate of change of in this relationship is , or . Substituting for in the equation yields . The value of can be found by substituting any value of and its corresponding value of for and , respectively, in this equation. Substituting for and for yields , or . Subtracting from both sides of this equation yields . Substituting for in the equation yields . Subtracting from both sides of this equation yields . Subtracting from both sides of this equation yields , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The y-intercept of the graph of in the xy-plane is . What is the value of ?
The correct answer is . It's given that the y-intercept of the graph of in the xy-plane is . Substituting for in the equation yields , or . Dividing both sides of this equation by yields . Therefore, the value of is .
Line is shown in the xy-plane. Line (not shown) is parallel to line . What is the slope of line ?
The correct answer is . It's given that line is parallel to line . It follows that the slope of line is equal to the slope of line . Given two points on a line in the xy-plane, and , the slope of the line can be calculated as . In the xy-plane shown, the points and are on line . It follows that the slope of line is , or . Since the slope of line is equal to the slope of line , the slope of line is also .
In the xy-plane, line passes through the point and is parallel to the line represented by the equation . If line also passes through the point , what is the value of ?
The correct answer is . A line in the xy-plane can be defined by the equation , where is the slope of the line and is the y-coordinate of the y-intercept of the line. It's given that line passes through the point . Therefore, the y-coordinate of the y-intercept of line is . It's given that line is parallel to the line represented by the equation . Since parallel lines have the same slope, it follows that the slope of line is . Therefore, line can be defined by an equation in the form , where and . Substituting for and for in yields the equation , or . If line passes through the point , then when , for the equation . Substituting for and for in the equation yields , or .
The function is defined by the given equation. What is the value of when ?
The correct answer is . It’s given that the function is defined by . Substituting for in the given function yields , which gives , or . Therefore, when , the value of is .
Brian saves of the he earns each week from his job. If Brian continues to save at this rate, how much money, in dollars, will Brian save in weeks?
The correct answer is . It's given that Brian saves of the he earns each week from his job. Therefore, Brian saves , or , per week. If Brian continues to save at this rate of per week for weeks, then he will save a total of , or , dollars.
What is the slope of the graph of in the xy-plane?
The correct answer is . A linear equation can be written in the form , where is the slope of the graph of the equation in the xy-plane and is the y-intercept. Distributing the in the equation yields . Combining like terms on the right-hand side of this equation yields . This equation is in the form , where and . Therefore, the slope of the graph of the given equation in the xy-plane is . Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.
One gallon of stain will cover square feet of a surface. A yard has a total fence area of square feet. Which equation represents the total amount of stain , in gallons, needed to stain the fence in this yard twice?
Choice D is correct. It's given that represents the total fence area, in square feet. Since the fence will be stained twice, the amount of stain, in gallons, will need to cover square feet. It’s also given that one gallon of stain will cover square feet. Dividing the total area, in square feet, of the surface to be stained by the number of square feet covered by one gallon of stain gives the number of gallons of stain that will be needed. Dividing by yields , or . Therefore, the equation that represents the total amount of stain , in gallons, needed to stain the fence of the yard twice is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
For the linear function , the graph of in the xy-plane passes through the points and . Which equation defines ?
Choice B is correct. It’s given that the graph of the linear function , where , passes through the points and in the xy-plane. An equation defining can be written in the form , where , represents the slope of the graph in the xy-plane, and represents the y-coordinate of the y-intercept of the graph. The slope can be found using any two points, and , and the formula . Substituting and for and , respectively, in the slope formula yields , which is equivalent to , or . Substituting for and for in the equation yields , or . Subtracting from each side of this equation yields . Substituting for and for in the equation yields . Since , it follows that the equation that defines is .
Choice A is incorrect. For this function, the graph of in the xy-plane would pass through , not , and , not .
Choice C is incorrect. For this function, the graph of in the xy-plane would pass through , not , and , not .
Choice D is incorrect. For this function, the graph of in the xy-plane would pass through , not , and , not .
The function gives the estimated height, in feet, of a willow tree years after its height was first measured. Which statement is the best interpretation of in this context?
The tree will be measured each year for years.
The tree is estimated to grow to a maximum height of feet.
The estimated height of the tree increased by feet each year.
The estimated height of the tree was feet when it was first measured.
Choice D is correct. It's given that the function gives the estimated height, in feet, of a willow tree years after its height was first measured. For a function defined by an equation of the form , where and are constants, represents the value of when . It follows that in the given function, represents the value of when . Therefore, the best interpretation of in this context is that the estimated height of the tree was feet when it was first measured.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Tom scored 85, 78, and 98 on his first three exams in history class. Solving which inequality gives the score, G, on Tom’s fourth exam that will result in a mean score on all four exams of at least 90 ?
Choice C is correct. The mean of the four scores (G, 85, 78, and 98) can be expressed as . The inequality that expresses the condition that the mean score is at least 90 can therefore be written as
.
Choice A is incorrect. The sum of the scores (G, 85, 78, and 98) isn’t divided by 4 to express the mean. Choice B is incorrect and may be the result of an algebraic error when multiplying both sides of the inequality by 4. Choice D is incorrect because it doesn’t include G in the mean with the other three scores.
In the system of equations below, a and c are constants.
If the system of equations has an infinite number of solutions , what is the value of a ?
0
Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side and a constant on the right-hand side of the equation. The coefficient of y in the second equation is equal to the coefficient of y in the first equation multiplied by 3. Therefore, a, the coefficient of x in the second equation, must be equal to 3 times the coefficient of x in the first equation: , or
.
Choices A, B, and C are incorrect. When ,
, or
, the given system of equations has one solution.
Sean rents a tent at a cost of per day plus a onetime insurance fee of . Which equation represents the total cost , in dollars, to rent the tent with insurance for days?
Choice C is correct. It’s given that the cost of renting a tent is per day for days. Multiplying the rental cost by the number of days yields , which represents the cost of renting the tent for days before the insurance is added. Adding the onetime insurance fee of to the rental cost of gives the total cost , in dollars, which can be represented by the equation .
Choice A is incorrect. This equation represents the total cost to rent the tent if the insurance fee was charged every day.
Choice B is incorrect. This equation represents the total cost to rent the tent if the daily fee was for days.
Choice D is incorrect. This equation represents the total cost to rent the tent if the daily fee was and the onetime fee was .
The graph of is shown. Which equation defines function ?
Choice A is correct. An equation for the graph shown can be written in slope-intercept form , where is the slope of the graph and its y-intercept is . Since the y-intercept of the graph shown is , the value of is . Since the graph also passes through the point , the slope can be calculated as , or . Therefore, the value of is . Substituting for and for in the equation yields . It’s given that an equation for the graph shown is . Substituting for in the equation yields . Subtracting from both sides of this equation yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Keenan made cups of vegetable broth. Keenan then filled small jars and large jars with all the vegetable broth he made. The equation represents this situation. Which is the best interpretation of in this context?
The number of large jars Keenan filled
The number of small jars Keenan filled
The total number of cups of vegetable broth in the large jars
The total number of cups of vegetable broth in the small jars
Choice C is correct. It’s given that the equation represents the situation where Keenan filled small jars and large jars with all the vegetable broth he made, which was cups. Therefore, represents the total number of cups of vegetable broth in the small jars and represents the total number of cups of vegetable broth in the large jars.
Choice A is incorrect. The number of large jars Keenan filled is represented by , not .
Choice B is incorrect. The number of small jars Keenan filled is represented by , not .
Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by , not .
During spring migration, a dragonfly traveled a minimum of miles and a maximum of miles between stopover locations. Which inequality represents this situation, where is a possible distance, in miles, this dragonfly traveled between stopover locations during spring migration?
Choice B is correct. It's given that during spring migration, a dragonfly traveled a minimum of miles and a maximum of miles between stopover locations. It's also given that represents a possible distance, in miles, this dragonfly traveled between stopover locations. It follows that the inequality represents this situation.
Choice A is incorrect. This inequality represents a situation in which a dragonfly traveled a maximum of miles between stopover locations.
Choice C is incorrect. This inequality represents a situation in which a dragonfly traveled a minimum of miles between stopover locations.
Choice D is incorrect. This inequality represents a situation in which a dragonfly traveled a minimum of miles and a maximum of miles between stopover locations.
The boiling point of water at sea level is 212 degrees Fahrenheit . For every 550 feet above sea level, the boiling point of water is lowered by about
. Which of the following equations can be used to find the boiling point B of water, in
, x feet above sea level?
Choice D is correct. It’s given that the boiling point of water at sea level is 212°F and that for every 550 feet above sea level, the boiling point of water is lowered by about 1°F. Therefore, the change in the boiling point of water x feet above sea level is represented by the expression . Adding this expression to the boiling point of water at sea level gives the equation for the boiling point B of water, in °F, x feet above sea level:
, or
.
Choices A and B are incorrect and may result from using the boiling point of water at sea level as the rate of change and the rate of change as the initial boiling point of water at sea level. Choice C is incorrect and may result from representing the change in the boiling point of water as an increase rather than a decrease.
Which point (, ) is a solution to the given inequality in the -plane?
(, )
(, )
(, )
(, )
Choice D is correct. For a point to be a solution to the given inequality in the xy-plane, the value of the point’s y-coordinate must be less than the value of , where is the value of the x-coordinate of the point. This is true of the point because , or . Therefore, the point is a solution to the given inequality.
Choices A, B, and C are incorrect. None of these points are a solution to the given inequality because each point’s y-coordinate is greater than the value of for the point’s x-coordinate.
Some values of the linear function f are shown in the table above. Which of the following defines f ?
Choice C is correct. Because f is a linear function of x, the equation , where m and b are constants, can be used to define the relationship between x and f (x). In this equation, m represents the increase in the value of f (x) for every increase in the value of x by 1. From the table, it can be determined that the value of f (x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is
, or 4. The value of b can be found by substituting the values of x and f (x) from any row of the table and the value of m into the equation
and solving for b. For example, using
,
, and
yields
. Solving for b yields
. Therefore, the equation defining the function f can be written in the form
.
Choices A, B, and D are incorrect. Any equation defining the linear function f must give values of f (x) for corresponding values of x, as shown in each row of the table. According to the table, if ,
. However, substituting
into the equation given in choice A gives
, or
, not 13. Similarly, substituting
into the equation given in choice B gives
, or
, not 13.
Lastly, substituting into the equation given in choice D gives
, or
, not 13. Therefore, the equations in choices A, B, and D cannot define f.
The function represents the total cost, in dollars, of attending an arcade when games are played. How many games can be played for a total cost of ?
The correct answer is . It’s given that the function represents the total cost, in dollars, of attending an arcade when games are played. Substituting for in the given equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, games can be played for a total cost of .
The graph shows the possible combinations of the number of pounds of tangerines and lemons that could be purchased for at a certain store. If Melvin purchased lemons and pounds of tangerines for a total of , how many pounds of lemons did he purchase?
Choice B is correct. It's given that the graph shows the possible combinations of the number of pounds of tangerines, , and the number of pounds of lemons, , that could be purchased for at a certain store. If Melvin purchased lemons and pounds of tangerines for a total of , the number of pounds of lemons he purchased is represented by the y-coordinate of the point on the graph where . For the graph shown, when , . Therefore, if Melvin purchased lemons and pounds of tangerines for a total of , then he purchased pounds of lemons.
Choice A is incorrect. This is the number of pounds of tangerines Melvin purchased if he purchased tangerines and pounds of lemons for a total of .
Choice C is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and pounds of tangerines for a total of .
Choice D is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and pound of tangerines for a total of .
A certain product costs a company $65 to make. The product is sold by a salesperson who earns a commission that is equal to 20% of the sales price of the product. The profit the company makes for each unit is equal to the sales price minus the combined cost of making the product and the commission. If the sales price of the product is $100, which of the following equations gives the number of units, u, of the product the company sold to make a profit of $6,840 ?
Choice A is correct. The sales price of one unit of the product is given as $100. Because the salesperson is awarded a commission equal to 20% of the sales price, the expression 100(1 – 0.2) gives the sales price of one unit after the commission is deducted. It is also given that the profit is equal to the sales price minus the combined cost of making the product, or $65, and the commission: 100(1 – 0.2) – 65. Multiplying this expression by u gives the profit of u units: (100(1 – 0.2) – 65)u. Finally, it is given that the profit for u units is $6,840; therefore (100(1 – 0.2) – 65)u = $6,840.
Choice B is incorrect. In this equation, cost is subtracted before commission and the equation gives the commission, not what the company retains after commission. Choice C is incorrect because the number of units is multiplied only by the cost but not by the sale price. Choice D is incorrect because the value 0.2 shows the commission, not what the company retains after commission.
A bus is traveling at a constant speed along a straight portion of road. The equation gives the distance , in feet from a road marker, that the bus will be seconds after passing the marker. How many feet from the marker will the bus be seconds after passing the marker?
Choice C is correct. It’s given that represents the number of seconds after the bus passes the marker. Substituting for in the given equation yields , or . Therefore, the bus will be feet from the marker seconds after passing it.
Choice A is incorrect. This is the distance, in feet, the bus will be from the marker second, not seconds, after passing it.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the distance, in feet, the bus will be from the marker seconds, not seconds, after passing it.
The solution to the given system of equations is . What is the value of ?
The correct answer is . The first equation in the given system of equations defines as . Substituting for in the second equation in the given system of equations yields . Applying the distributive property on the left-hand side of this equation yields . Subtracting from each side of this equation yields . Subtracting from each side of this equation yields . Substituting for in the first equation of the given system of equations yields , or . Substituting for and for into the expression yields , or .
The cost , in dollars, for a manufacturer to make rings is represented by the line shown.
What is the cost, in dollars, for the manufacturer to make rings?
Choice C is correct. The line shown represents the cost , in dollars, for a manufacturer to make rings. For the line shown, the x-axis represents the number of rings made by the manufacturer and the y-axis represents the cost, in dollars. Therefore, the cost, in dollars, for the manufacturer to make rings is represented by the y-coordinate of the point on the line that has an x-coordinate of . The point on the line with an x-coordinate of has a y-coordinate of . Therefore, the cost, in dollars, for the manufacturer to make rings is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the cost, in dollars, for the manufacturer to make rings.
Choice D is incorrect. This is the cost, in dollars, for the manufacturer to make rings.
more than times a number is equal to . Which equation represents this situation?
Choice D is correct. The given phrase “ times a number ” can be represented by the expression . The given phrase “ more than” indicates an increase of to a quantity. Therefore “ more than times a number ” can be represented by the expression . Since it’s given that more than times a number is equal to , it follows that is equal to , or . Therefore, the equation that represents this situation is .
Choice A is incorrect. This equation represents times the quantity times a number is equal to .
Choice B is incorrect. This equation represents times a number is equal to more than .
Choice C is incorrect. This equation represents more than times a number is equal to .
If is the solution to the system of equations above, what is the value of
?
5
13
Choice B is correct. Subtracting the second equation, , from the first equation,
, results in
, or
. Combining like terms on the left-hand side of this equation yields
.
Choice A is incorrect and may result from miscalculating as
. Choice C is incorrect and may result from miscalculating
as 5. Choice D is incorrect and may result from adding 9 to 4 instead of subtracting 9 from 4.
What is an equation of the graph shown?
Choice C is correct. An equation of a line can be written in the form , where is the slope of the line and is the y-intercept of the line. The line shown passes through the point , so . The line shown also passes through the point . The slope, , of a line passing through two points and can be calculated using the equation . For the points and , this gives , or . Substituting for and for in yields , or . Therefore, an equation of the graph shown is .
Choice A is incorrect. This is an equation of a line with a slope of , not .
Choice B is incorrect. This is an equation of a line with a slope of , not .
Choice D is incorrect. This is an equation of a line with a slope of , not .
The function is defined by . What is the value of ?
Choice B is correct. It’s given that the function is defined by . The value of is the value of when . Substituting for in the given equation yields , which is equivalent to , or .
Choice A is incorrect. This is the value of when , rather than .
Choice C is incorrect. This is the value of , rather than .
Choice D is incorrect. This is the value of , rather than .
A city’s total expense budget for one year was x million dollars. The city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. Which of the following represents the relationship between x and y in this context?
Choice B is correct. Of the city’s total expense budget for one year, the city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. This means that the expression represents the total expense budget, in millions of dollars, for one year. It’s given that the total expense budget for one year is x million dollars. It follows then that the expression
is equivalent to x, or
. Subtracting y from both sides of this equation yields
. By the symmetric property of equality, this is the same as
.
Choices A and C are incorrect. Because it’s given that the total expense budget for one year, x million dollars, is comprised of the departmental expenses, y million dollars, and all other expenses, 201 million dollars, the expressions and
both must be equivalent to a value greater than 201 million dollars. Therefore, the equations
and
aren’t true. Choice D is incorrect. The value of x must be greater than the value of y. Therefore,
can’t represent this relationship.
The solution to the given system of equations is . What is the value of ?
Choice B is correct. It's given by the first equation in the system that . Substituting for in the second equation in the system, , yields . Adding to both sides of this equation yields , which is equivalent to , or . Multiplying both sides of this equation by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. It's given that and is a system of equations with a solution . Adding the second equation in the given system to the first equation yields , which is equivalent to . Thus, the value of is .
Choice A is incorrect. This represents the value of .
Choice B is incorrect. This represents the value of .
Choice D is incorrect. This represents the value of .
The graph of the linear function is shown, where . What is the y-intercept of the graph of ?
Choice D is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function shown intersects the y-axis at the point . Therefore, the y-intercept of the graph of is .
Choice A is incorrect. This is the point where the x-axis, not the graph of , intersects the y-axis.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. Which of the following inequalities represents all possible weights w, in pounds, for the elephant 365 days after birth?
Choice D is correct. It’s given that the elephant weighs 200 pounds at birth and gains more than 2 pounds but less than 3 pounds per day during its first year. The inequality represents this situation, where d is the number of days after birth. Substituting 365 for d in the inequality gives
, or
.
Choice A is incorrect and may result from solving the inequality . Choice B is incorrect and may result from solving the inequality for a weight range of more than 1 pound but less than 2 pounds:
. Choice C is incorrect and may result from calculating the possible weight gained by the elephant during the first year without adding the 200 pounds the elephant weighed at birth.
For the function f defined above, what is the value of ?
1
2
Choice B is correct. The value of can be calculated by finding the values of
and
. The value of
can be found by substituting 9 for x in the given function:
. This equation can be rewritten as
, or 4. Then, the value of
can be found by substituting 1 for x in the given function:
. This equation can be rewritten as
, or 2. Therefore,
, which is equivalent to 2.
Choices A, C, and D are incorrect and may result from incorrectly substituting values of x in the given function or making computational errors.
In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?
and
and
and
and
Choice B is correct. Distributing the a on the left-hand side of the equation gives 3a – b – ax = –1 – 2x. Rearranging the terms in each side of the equation yields –ax + 3a – b = –2x –1. Since the equation has infinitely many solutions, it follows that the coefficients of x and the free terms on both sides must be equal. That is, –a = –2, or a = 2, and 3a – b = –1. Substituting 2 for a in the equation 3a – b = –1 gives 3(2) – b = –1, so b = 7.
Choice A is incorrect and may be the result of a conceptual error when finding the value of b. Choices C and D are incorrect and may result from making a sign error when simplifying.
A sample of a certain alloy has a total mass of grams and is silicon by mass. The sample was created by combining two pieces of different alloys. The first piece was silicon by mass and the second piece was silicon by mass. What was the mass, in grams, of the silicon in the second piece?
Choice B is correct. Let represent the total mass, in grams, of the first piece, and let represent the total mass, in grams, of the second piece. It's given that the sample has a total mass of grams. Therefore, the equation represents this situation. It's also given that the sample is silicon by mass. Therefore, the total mass of the silicon in the sample is , or , grams. It's also given that the first piece was silicon by mass and the second piece was silicon by mass. Therefore, the masses, in grams, of the silicon in the first and second pieces can be represented by the expressions and , respectively. Since the sample was created by combining the first and second pieces, and the total mass of the silicon in the sample is grams, the equation represents this situation. Subtracting from both sides of the equation yields . Substituting for in the equation yields . Distributing on the left-hand side of this equation yields . Combining like terms on the left-hand side of this equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the expression representing the mass, in grams, of the silicon in the second piece, , yields , or . Therefore, the mass, in grams, of the silicon in the second piece is .
Choice A is incorrect. This is the mass, in grams, of the silicon in the first piece, not the second piece.
Choice C is incorrect. This is the total mass, in grams, of the second piece, not the mass, in grams, of the silicon in the second piece.
Choice D is incorrect. This is the total mass, in grams, of the first piece, not the mass, in grams, of the silicon in the second piece.
A total of people contributed to a charity event as either a donor or a volunteer. people contributed as a donor. How many people contributed as a volunteer?
Choice A is correct. It’s given that a total of people contributed to a charity event as either a donor or a volunteer. It’s also given that people contributed as a donor. It follows that , or , people contributed as a volunteer.
Choice B is incorrect. This is the number of people who contributed as a donor, not a volunteer.
Choice C is incorrect. This is the total number of people who contributed as either a donor or a volunteer, not the number of people who contributed as a volunteer.
Choice D is incorrect and may result from conceptual or calculation errors.
Which equation has the same solution as the given equation?
Choice C is correct. Subtracting from both sides of the given equation yields , which is the equation given in choice C. Since this equation is equivalent to the given equation, it has the same solution as the given equation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
An event planner is planning a party. It costs the event planner a onetime fee of to rent the venue and per attendee. The event planner has a budget of . What is the greatest number of attendees possible without exceeding the budget?
The correct answer is . The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let be the number of attendees. The expression thus represents the total cost of the party. It's given that the budget is , so this situation can be represented by the inequality . Subtracting from both sides of this inequality gives . Dividing both sides of this inequality by results in approximately . Since the question is stated in terms of attendees, rounding down to the greatest whole number gives the greatest number of attendees possible, which is .
For a party, dinner rolls are needed. Dinner rolls are sold in packages of . What is the minimum number of packages that should be bought for the party?
The correct answer is . Let represent the number of packages of dinner rolls that should be bought for the party. It's given that dinner rolls are sold in packages of . Therefore, represents the number of dinner rolls that should be bought for the party. It's also given that dinner rolls are needed; therefore, . Dividing both sides of this inequality by yields , or approximately . Since the number of packages of dinner rolls must be a whole number, the minimum number of packages that should be bought for the party is .
In the equation above, a and b are constants and . Which of the following could represent the graph of the equation in the xy-plane?
Choice C is correct. The given equation can be rewritten in slope-intercept form,
, where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields
, and dividing both sides of this equation by b yields
, or
. With the equation now in slope-intercept form, it shows that
, which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that
. Since
, the slope of the line must be a value between
and 0. Choice C is the only graph of a line that has a y-value of the y-intercept that is 1 and a slope that is between
and 0.
Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between and 0.
The Karvonen formula above shows the relationship between Alice’s target heart rate H, in beats per minute (bpm), and the intensity level p of different activities. When , Alice has a resting heart rate. When
, Alice has her maximum heart rate. It is recommended that p be between 0.5 and 0.85 for Alice when she trains. Which of the following inequalities describes Alice’s target training heart rate?
Choice A is correct. When Alice trains, it’s recommended that p be between 0.5 and 0.85. Therefore, her target training heart rate is represented by the values of H corresponding to . When
,
, or
. When
,
, or
. Therefore, the inequality that describes Alice’s target training heart rate is
.
Choice B is incorrect. This inequality describes Alice’s target heart rate for . Choice C is incorrect. This inequality describes her target heart rate for
. Choice D is incorrect. This inequality describes her target heart rate for
.
The function is defined by . What is the value of ?
Choice A is correct. The value of can be found by substituting for in the equation defining . Substituting for in yields , or . Therefore, the value of is .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given equation, and are constants and . The equation has no solution. What is the value of ?
The correct answer is . A linear equation in the form has no solution only when the coefficients of on each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by yields , or . Since it’s given that the equation has no solution, the coefficient of on both sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since , and it's given that , the second condition is true. Thus, must be equal to . Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer.
Which table gives three values of and their corresponding values of for the given equation?
Choice C is correct. Each of the given choices gives three values of : , , and . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Choice C gives three values of , , , and , and their corresponding values of , , , and , respectively, for the given equation.
Choice A is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice B is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice D is incorrect. This table gives three values of and their corresponding values of for the equation .
An agricultural scientist studying the growth of corn plants recorded the height of a corn plant at the beginning of a study and the height of the plant each day for the next 12 days. The scientist found that the height of the plant increased by an average of 1.20 centimeters per day for the 12 days. If the height of the plant on the last day of the study was 36.8 centimeters, what was the height, in centimeters, of the corn plant at the beginning of the study?
The correct answer is 22.4. If the height of the plant increased by an average of 1.20 centimeters per day for 12 days, then its total growth over the 12 days was centimeters. The plant was 36.8 centimeters tall after 12 days, so at the beginning of the study its height was
centimeters. Note that 22.4 and 112/5 are examples of ways to enter a correct answer.
Alternate approach: The equation can be used to represent this situation, where h is the height of the plant, in centimeters, at the beginning of the study. Solving this equation for h yields 22.4 centimeters.
For the linear function , the table shows three values of and their corresponding values of . Which equation defines ?
Choice A is correct. An equation that defines a linear function can be written in the form , where and are constants. It's given in the table that when , . Substituting for and for in the equation yields , or . Substituting for in the equation yields . It's also given in the table that when , . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Substituting for in the equation yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The points plotted in the coordinate plane above represent the possible numbers of wallflowers and cornflowers that someone can buy at the Garden Store in order to spend exactly $24.00 total on the two types of flowers. The price of each wallflower is the same and the price of each cornflower is the same. What is the price, in dollars, of 1 cornflower?
The correct answer is 1.5. The point corresponds to the situation where 16 cornflowers and 0 wallflowers are purchased. Since the total spent on the two types of flowers is $24.00, it follows that the price of 16 cornflowers is $24.00, and the price of one cornflower is $1.50. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.
A small business owner budgets to purchase candles. The owner must purchase a minimum of candles to maintain the discounted pricing. If the owner pays per candle to purchase small candles and per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?
The correct answer is . Let represent the number of small candles the owner can purchase, and let represent the number of large candles the owner can purchase. It’s given that the owner pays per candle to purchase small candles and per candle to purchase large candles. Therefore, the owner pays dollars for small candles and dollars for large candles, which means the owner pays a total of dollars to purchase candles. It’s given that the owner budgets to purchase candles. Therefore, . It’s also given that the owner must purchase a minimum of candles. Therefore, . The inequalities and can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting from both sides of the inequality yields . Multiplying both sides of this inequality by yields , or . Adding to both sides of this inequality yields , or . This inequality can be combined with the inequality , which yields the compound inequality . It follows that . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields approximately . Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than , which is .
What value of is the solution to the given equation?
The correct answer is . The expression is a factor of both terms on the left-hand side of the given equation. Therefore, the given equation can be written as , or . Multiplying each side of this equation by yields . Subtracting from each side of this equation yields . Therefore, the value of that is the solution to the given equation is .
A number is at most less than times the value of . If the value of is , what is the greatest possible value of ?
The correct answer is . It's given that a number is at most less than times the value of , or . Substituting for in this inequality yields , or . Thus, if the value of is , the greatest possible value of is .
A food truck buys forks for each and plates for each. The total cost of forks and plates is . Which equation represents this situation?
Choice D is correct. It’s given that the food truck buys forks for each. Therefore, the cost, in dollars, of forks can be represented by the expression . It’s also given that the food truck buys plates for each. Therefore, the cost, in dollars, of plates can be represented by the expression . Since the total cost of forks and plates is , the equation represents this situation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This equation represents a situation in which the food truck buys forks for each and plates for each.
The table shows three values of and their corresponding values of , where is a constant. There is a linear relationship between and . Which of the following equations represents this relationship?
Choice B is correct. The linear relationship between and can be represented by an equation of the form , where is the slope of the graph of the equation in the xy-plane and is a point on the graph. The slope of a line can be found using two points on the line and the slope formula . Each value of and its corresponding value of in the table can be represented by a point . Substituting the points and for and , respectively, in the slope formula yields , which gives , or . Substituting for and the point for in the equation yields . Distributing on the right-hand side of this equation yields . Adding to each side of this equation yields . Multiplying each side of this equation by yields . Adding to each side of this equation yields . Therefore, the equation represents this relationship.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph of in the -plane has an -intercept at and a -intercept at , where and are constants. What is the value of ?
Choice D is correct. The x-coordinate of the x-intercept can be found by substituting for in the given equation, which gives , or . Dividing both sides of this equation by yields . Therefore, the value of is . The y-coordinate of the y-intercept can be found by substituting for in the given equation, which gives , or . Dividing both sides of this equation by yields . Therefore, the value of is . It follows that the value of is , which is equivalent to , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Among the remaining 150 participants, p people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of p ?
, where
, where
, where
, where
Choice D is correct. Of the first 150 participants, 36 chose the first picture in the set, and of the 150 remaining participants, p chose the first picture in the set. Hence, the proportion of the participants who chose the first picture in the set is . Since more than 20% of all the participants chose the first picture, it follows that
.
This inequality can be rewritten as . Since p is a number of people among the remaining 150 participants,
.
Choices A, B, and C are incorrect and may be the result of some incorrect interpretations of the given information or of computational errors.
The score on a trivia game is obtained by subtracting the number of incorrect answers from twice the number of correct answers. If a player answered 40 questions and obtained a score of 50, how many questions did the player answer correctly?
The correct answer is 30. Let x represent the number of correct answers from the player and y represent the number of incorrect answers from the player. Since the player answered 40 questions in total, the equation represents this situation. Also, since the score is found by subtracting the number of incorrect answers from twice the number of correct answers and the player received a score of 50, the equation
represents this situation. Adding the equations in the system of two equations together yields
. This can be rewritten as
. Finally, solving for x by dividing both sides of the equation by 3 yields
.
Nasir bought storage bins that were each the same price. He used a coupon for off the entire purchase. The cost for the entire purchase after using the coupon was . What was the original price, in dollars, for storage bin?
The correct answer is . It’s given that the cost for the entire purchase was after a coupon was used for off the entire purchase. Adding the amount of the coupon to the purchase price yields . Thus, the cost for the entire purchase before using the coupon was . It’s given that Nasir bought storage bins. The original price for storage bin can be found by dividing the total cost by . Therefore, the original price, in dollars, for storage bin is , or .
The graph shows the relationship between the number of shares of stock from Company A, , and the number of shares of stock from Company B, , that Simone can purchase. Which equation could represent this relationship?
Choice B is correct. The graph shown is a line passing through the points and . Since the relationship between and is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting for and for in the equation in choice B, , yields , or , which is true. Substituting for and for in the equation yields , or , which is true. Therefore, the equation represents the relationship between and .
Choice A is incorrect. The point is not on the graph of this equation, since , or , is not true.
Choice C is incorrect. The point is not on the graph of this equation, since , or , is not true.
Choice D is incorrect. The point is not on the graph of this equation, since , or , is not true.
Mario purchased 4 binders that cost x dollars each and 3 notebooks that cost y dollars each. If the given equation represents this situation, which of the following is the best interpretation of 24 in this context?
The total cost, in dollars, for all binders purchased
The total cost, in dollars, for all notebooks purchased
The total cost, in dollars, for all binders and notebooks purchased
The difference in the total cost, in dollars, between the number of binders and notebooks purchased
Choice C is correct. Since Mario purchased 4 binders that cost x dollars each, the expression represents the total cost, in dollars, of the 4 binders he purchased. Since Mario purchased 3 notebooks that cost y dollars each, the expression
represents the total cost, in dollars, of the 3 notebooks he purchased. Therefore, the expression
represents the total cost, in dollars, for all binders and notebooks he purchased. In the given equation, the expression
is equal to 24. Therefore, it follows that 24 is the total cost, in dollars, for all binders and notebooks purchased.
Choice A is incorrect. This is represented by the expression in the given equation. Choice B is incorrect. This is represented by the expression
in the given equation. Choice D is incorrect. This is represented by the expression
.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form , where , , and are constants, are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in standard form by subtracting from both sides of the equation to yield . The second equation in the given system can be written in standard form by adding to both sides of the equation to yield . The coefficient of in this second equation, , is times the coefficient of in the first equation, . For the lines to be parallel the coefficient of in the second equation, , must also be times the coefficient of in the first equation, . Thus, , or . Therefore, if the given system has no solution, the value of is .
How many solutions exist to the equation shown above?
None
Exactly 1
Exactly 3
Infinitely many
Choice B is correct. Subtracting 4 from each side of the given equation yields , or
, so the equation has a unique solution of
.
Choice A is incorrect. Since 3 is a value of x that satisfies the given equation, the equation has at least 1 solution. Choice C is incorrect. Linear equations can have 0, 1, or infinitely many solutions; no linear equation has exactly 3 solutions. Choice D is incorrect. If a linear equation has infinitely many solutions, it can be reduced to . This equation reduces to
, so there is only 1 solution.
For a training program, Juan rides his bike at an average rate of minutes per mile. Which function models the number of minutes it will take Juan to ride miles at this rate?
Choice D is correct. It′s given that Juan rides his bike at an average rate of minutes per mile. The number of minutes it will take Juan to ride miles can be determined by multiplying his average rate by the number of miles, , which yields . Therefore, the function models the number of minutes it will take Juan to ride miles.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
For the linear function , the table shows three values of and their corresponding values of . Function is defined by , where and are constants. What is the value of ?
Choice D is correct. The table gives that when . Substituting 0 for and for into the equation yields . Subtracting from both sides of this equation yields . The table gives that when . Substituting for , for , and for into the equation yields . Combining like terms yields , or . Since , substituting for into this equation gives , which yields . Thus, the value of can be written as , which is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
What value of is the solution to the given equation?
The correct answer is . Subtracting from both sides of the given equation yields . Therefore, the value of is .
What is the solution to the given system of equations?
Choice C is correct. Adding to both sides of the first equation in the given system yields . Substituting the expression for in the second equation in the given system yields . Distributing the on the right-hand side of this equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields , or . Therefore, the solution to the given system of equations is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the solution , not , to the given system of equations.
What is the solution to the given system of equations?
Choice B is correct. It's given by the second equation in the system that . Substituting for in the first equation yields . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, since and , the solution to the given system of equations is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. Assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000?
Choice C is correct. It is assumed that the oil and gas production decreased at a constant rate. Therefore, the function f that best models the production t years after the year 2000 can be written as a linear function, , where m is the rate of change of the oil and gas production and b is the oil and gas production, in millions of barrels, in the year 2000. Since there were 4 million barrels of oil and gas produced in 2000,
. The rate of change, m, can be calculated as
, which is equivalent to
, the rate of change in choice C.
Choices A and B are incorrect because each of these functions has a positive rate of change. Since the oil and gas production decreased over time, the rate of change must be negative. Choice D is incorrect. This model may result from misinterpreting 1.9 million barrels as the amount by which the production decreased.
Some values of the linear function f are shown in the table above. What is the value of ?
6
7
8
9
Choice B is correct. A linear function has a constant rate of change, and any two rows of the table shown can be used to calculate this rate. From the first row to the second, the value of x is increased by 2 and the value of is increased by
. So the values of
increase by 3 for every increase by 1 in the value of x. Since
, it follows that
. Therefore,
.
Choice A is incorrect. This is the third x-value in the table, not . Choices C and D are incorrect and may result from errors when calculating the function’s rate of change.
For which of the following tables are all the values of and their corresponding values of solutions to the given system of inequalities?
Choice A is correct. The inequality indicates that for any solution to the given system of inequalities, the value of must be greater than the corresponding value of . The inequality indicates that for any solution to the given system of inequalities, the value of must be less than . Of the given choices, only choice A contains values of that are each greater than the corresponding value of and less than . Therefore, for choice A, all the values of and their corresponding values of are solutions to the given system of inequalities.
Choice B is incorrect. The values in this table aren’t solutions to the inequality .
Choice C is incorrect. The values in this table aren’t solutions to the inequality .
Choice D is incorrect. The values in this table aren’t solutions to the inequality or the inequality .
A line segment that has a length of is divided into three parts. One part is long. The other two parts have lengths that are equal to each other. What is the length, in , of one of the other two parts of equal length?
The correct answer is . It’s given that a line segment has a length of and is divided into three parts, where one part is long and the other two parts have lengths that are equal. If represents the length, in cm, of each of the two parts of equal length, then the equation , or , represents this situation. Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, the length, in cm, of one of the two parts of equal length is .
The solution to the given system of equations is . What is the value of ?
The correct answer is . Subtracting the second equation in the given system from the first equation yields , which is equivalent to , or . Dividing each side of this equation by yields . Substituting for in the second equation yields . Adding to both sides of this equation yields .
Alternate approach: Multiplying each side of the second equation in the given system by yields . Subtracting the first equation in the given system from this equation yields , which is equivalent to , or . Dividing each side of this equation by yields .
If , what is the value of ?
The correct answer is . Dividing both sides of the given equation by yields . Multiplying both sides of this equation by yields . Thus, the value of is .
In the xy-plane, the points and
lie on the graph of which of the following linear functions?
Choice C is correct. A linear function can be written in the form , where m is the slope and b is the y-coordinate of the y-intercept of the line. The slope of the graph can be found using the formula
. Substituting the values of the given points into this formula yields
or
, which simplifies to
. Only choice C shows an equation with this slope.
Choices A, B, and D are incorrect and may result from computation errors or misinterpreting the given information.
The function is defined by . What is the value of ?
Choice D is correct. It’s given that the function is defined by . Substituting for in the given function yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater sold 30 tickets for a total of $300, how much, in dollars, was spent on adult tickets? (Disregard the $ sign when gridding your answer.)
The correct answer is 180. Let a be the number of adult tickets sold and c be the number of child tickets sold. Since the theater sold a total of 30 tickets, a + c = 30. The price per adult ticket is $12, and the price per child ticket is $8. Since the theater received a total of $300 for the 30 tickets sold, it follows that 12a + 8c = 300. To eliminate c, the first equation can be multiplied by 8 and then subtracted from the second equation:
Because the question asks for the amount spent on adult tickets, which is 12a dollars, the resulting equation can be multiplied by 3 to give 3(4a) = 3(60) = 180. Therefore, $180 was spent on adult tickets.
Alternate approach: If all the 30 tickets sold were child tickets, their total price would be 30($8) = $240. Since the actual total price of the 30 tickets was $300, the extra $60 indicates that a certain number of adult tickets, a, were sold. Since the price of each adult ticket is $4 more than each child ticket, 4a = 60, and it follows that 12a = 180.
The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after the product is placed on the belt. Which of the following equations correctly relates d and m ?
Choice A is correct. The line passes through the origin. Therefore, this is a relationship of the form , where k is a constant representing the slope of the graph. To find the value of k, choose a point
on the graph of the line other than the origin and substitute the values of m and d into the equation. For example, if the point
is chosen, then
, and
. Therefore, the equation of the line is
.
Choice B is incorrect and may result from calculating the slope of the line as the change in time over the change in distance traveled instead of the change in distance traveled over the change in time. Choices C and D are incorrect because each of these equations represents a line with a d-intercept of 2. However, the graph shows a line with a d-intercept of 0.
What value of is the solution to the given equation?
The correct answer is . The given equation can be rewritten as . Dividing both sides of this equation by yields . Subtracting from both sides of this equation yields . Therefore, is the value of that is the solution to the given equation.
How many solutions does the equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice A is correct. Distributing on the left-hand side and on the right-hand side of the given equation yields . Adding to each side of this equation yields . Adding to each side of this equation yields . Dividing each side of this equation by yields . This means that is the only solution to the given equation. Therefore, the given equation has exactly one solution.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of ?
Choice A is correct. It’s given that the function is defined by . Substituting for into the given equation yields , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of for which , not the value of .
A producer is creating a video with a length of minutes. The video will consist of segments that are minute long and segments that are minutes long. Which equation represents this situation, where represents the number of -minute segments and represents the number of -minute segments?
Choice D is correct. Since represents the number of -minute segments and represents the number of -minute segments, the total length of the video is , or , minutes. Since the video is minutes long, the equation represents this situation.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
A company’s total cost , in dollars, to produce x shirts is given by the function above, where m is a constant and
. The total cost to produce 100 shirts is $800. What is the total cost, in dollars, to produce 1000 shirts? (Disregard the $ sign when gridding your answer.)
The correct answer is 3500. The given information includes a cost, $800, to produce 100 shirts. Substituting and
into the given equation yields
. Subtracting 500 from both sides of the equation yields
. Dividing both sides of this equation by 100 yields
. Substituting the value of m into the given equation yields
. Substituting 1000 for x in this equation and solving for
gives the cost of 1000 shirts:
, or 3500.
For the linear function , the graph of in the xy-plane has a slope of and passes through the point . Which equation defines ?
Choice D is correct. An equation defining a linear function can be written in the form , where is the slope and is the y-intercept of the graph of in the xy-plane. It’s given that the graph of has a slope of , so . It’s also given that the graph of passes through the point , so . Substituting for and for in yields , or . Thus, the equation that defines is .
Choice A is incorrect. This equation defines a function whose graph has a slope of , not .
Choice B is incorrect. This equation defines a function whose graph has a slope of , not .
Choice C is incorrect. This equation defines a function whose graph has a slope of , not , and passes through the point , not .
A gardener buys two kinds of fertilizer. Fertilizer A contains 60% filler materials by weight and Fertilizer B contains 40% filler materials by weight. Together, the fertilizers bought by the gardener contain a total of 240 pounds of filler materials. Which equation models this relationship, where x is the number of pounds of Fertilizer A and y is the number of pounds of Fertilizer B?
Choice B is correct. Since Fertilizer A contains 60% filler materials by weight, it follows that x pounds of Fertilizer A consists of 0.6x pounds of filler materials. Similarly, y pounds of Fertilizer B consists of 0.4y pounds of filler materials. When x pounds of Fertilizer A and y pounds of Fertilizer B are combined, the result is 240 pounds of filler materials. Therefore, the total amount, in pounds, of filler materials in a mixture of x pounds of Fertilizer A and y pounds of Fertilizer B can be expressed as .
Choice A is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B. Fertilizer A consists of 0.6x pounds of filler materials and Fertilizer B consists of 0.4y pounds of filler materials. Therefore, is equal to 240, not
. Choice C is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B and incorrectly represents how to take the percentage of a value mathematically. Choice D is incorrect. This choice incorrectly represents how to take the percentage of a value mathematically. Fertilizer A consists of 0.6x pounds of filler materials, not 60x pounds of filler materials, and Fertilizer B consists of 0.4y pounds of filler materials, not 40y pounds of filler materials.
One gallon of paint will cover square feet of a surface. A room has a total wall area of square feet. Which equation represents the total amount of paint , in gallons, needed to paint the walls of the room twice?
Choice A is correct. It's given that represents the total wall area, in square feet. Since the walls of the room will be painted twice, the amount of paint, in gallons, needs to cover square feet. It’s also given that one gallon of paint will cover square feet. Dividing the total area, in square feet, of the surface to be painted by the number of square feet covered by one gallon of paint gives the number of gallons of paint that will be needed. Dividing by yields , or . Therefore, the equation that represents the total amount of paint , in gallons, needed to paint the walls of the room twice is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from finding the amount of paint needed to paint the walls once rather than twice.
Choice D is incorrect and may result from conceptual or calculation errors.
The shaded region shown represents the solutions to which inequality?
Choice D is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form , where is the slope and is the y-intercept of the line. For the graph shown, the boundary line passes through the points and . Given two points on a line, and , the slope of the line can be calculated using the equation . Substituting the points and for and in this equation yields , which is equivalent to , or . Since the point represents the y-intercept, it follows that . Substituting for and for in the equation yields as the equation of the boundary line. Since the shaded region represents all the points above this boundary line, it follows that the shaded region shown represents the solutions to the inequality .
Choice A is incorrect. This inequality represents a region below, not above, a boundary line with a slope of , not .
Choice B is incorrect. This inequality represents a region below, not above, the boundary line shown.
Choice C is incorrect. This inequality represents a region whose boundary line has a slope of , not .
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice A is correct. Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . For the table in choice A, when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than ; and when , the corresponding value of is , which is less than . Therefore, the table in choice A gives values of and their corresponding values of that are all solutions to the given inequality.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A geologist needs to collect at least samples of lava from a volcano. If the geologist has already collected samples from the volcano, what is the minimum number of additional samples the geologist needs to collect?
Choice C is correct. It's given that the geologist has already collected samples from the volcano. Let represent the number of additional samples the geologist needs to collect. After collecting additional samples, the geologist will have collected a total of samples. It's given that the geologist needs to collect at least samples. Therefore, . Subtracting from each side of this inequality yields the inequality . Thus, the geologist needs to collect a minimum of additional samples.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the number of samples the geologist has already collected, rather than the minimum number of additional samples the geologist needs to collect.
Choice D is incorrect. If the geologist collects additional samples, the geologist will have collected a total of samples, which is less than samples.
For the function , the graph of in the xy-plane has a slope of and passes through the point . Which equation defines ?
Choice B is correct. An equation defining a linear function can be written in the form , where and are constants, is the slope of the graph of in the xy-plane, and is the y-intercept of the graph. It's given that for the function , the graph of in the xy-plane has a slope of . Therefore, . It's also given that this graph passes through the point . Therefore, the y-intercept of the graph is , and it follows that . Substituting for and for in the equation yields . Thus, the equation that defines is .
Choice A is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
Choice C is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
Choice D is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
In the xy-plane, a line has a slope of 6 and passes through the point . Which of the following is an equation of this line?
Choice A is correct. The slope-intercept form of an equation for a line is , where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It’s given that the slope is 6, so
. It’s also given that the line passes through the point
on the y-axis, so
. Substituting
and
into the equation
gives
.
Choices B, C, and D are incorrect and may result from misinterpreting the slope-intercept form of an equation of a line.
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of and , which inequality represents the possible lengths, , of the third side of the triangle?
or
Choice C is correct. It’s given that a triangle has side lengths of and , and represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities , , and represent all possible values of . Subtracting from both sides of the inequality yields , or . Adding and in the inequality yields , or . Subtracting from both sides of the inequality yields , or . Since all x-values that satisfy the inequality also satisfy the inequality , it follows that the inequalities and represent the possible values of . Therefore, the inequality represents the possible lengths, , of the third side of the triangle.
Choice A is incorrect. This inequality gives the upper bound for but does not include its lower bound.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of ?
The correct answer is . It’s given that . Substituting for in the second equation in the system, , yields , which gives , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Substituting for in the first equation in the system, , yields , or . Therefore, the value of is .
If , what is the value of ?
Choice C is correct. Subtracting from both sides of the given equation yields , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not .
Choice D is incorrect and may result from conceptual or calculation errors.
An artist paints and sells square tiles. The selling price P, in dollars, of a painted tile is a linear function of the side length of the tile s, in inches, as shown in the table below.
| Side length, s (inches) | Price, P (dollars) |
| 3 | 8.00 |
| 6 | 18.00 |
| 9 | 28.00 |
Which of the following could define the relationship between s and P ?
Choice C is correct. The relationship between s and P can be modeled by a linear equation of the form P = ks + a, where k and a are constants. The table shows that P increases by 10 when s increases by 3, so k = . To solve for a, substitute one of the given pairs of values for s and P: when s = 3, P = 8, so
, which yields a = –2. The solution is therefore
.
Choice A is incorrect. When s = 3, P = 8, but 3(3) + 10 = 19 ≠︀ 8. Choice B is incorrect. This may result from using the first number given for P in the table as the constant term a in the linear equation P = ks + a, which is true only when s = 0. Choice D is incorrect and may result from using the reciprocal of the slope of the line.
The function gives the estimated surface water temperature , in degrees Celsius, of a body of water on the day of the year, where . Based on the model, what is the estimated surface water temperature, in degrees Celsius, of this body of water on the day of the year?
Choice B is correct. It’s given that the function gives the estimated surface water temperature, in degrees Celsius, of a body of water on the day of the year. Substituting for in the given function yields , which is equivalent to , or . Therefore, the estimated surface water temperature, in degrees Celsius, of this body of water on the day of the year is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the rate of change, in degrees Celsius per day, of the estimated surface water temperature.
Choice D is incorrect. This is the change, in degrees Celsius, in the estimated surface water temperature over days.
Cathy has n CDs. Gerry has 3 more than twice the number of CDs that Cathy has. In terms of n, how many CDs does Gerry have?
Choice D is correct. The term 2n represents twice the number of CDs that Cathy has, and adding 3 represents 3 more than that amount.
Choices A and B are incorrect. The expression 3n represents three times the number of CDs that Cathy has. Choice C is incorrect. Subtracting 3 represents 3 fewer than twice the number of CDs that Cathy has.
The equation represents the number of sweaters, , and number of shirts, , that Yesenia purchased for . If Yesenia purchased sweaters, how many shirts did she purchase?
Choice B is correct. It's given that the equation represents the number of sweaters, , and the number of shirts, , that Yesenia purchased for . If Yesenia purchased sweaters, the number of shirts she purchased can be calculated by substituting for in the given equation, which yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, if Yesenia purchased sweaters, she purchased shirts.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the number of shirts Yesenia purchased if she purchased sweaters.
Choice D is incorrect. This is the price, in dollars, for each sweater, not the number of shirts Yesenia purchased.
Marisol drove 3 hours from City A to City B. The equation below estimates the distance d, in miles, Marisol traveled after driving for t hours.
Which of the following does 45 represent in the equation?
Marisol took 45 trips from City A to City B.
The distance between City A and City B is 45 miles.
Marisol drove at an average speed of about 45 miles per hour.
It took Marisol 45 hours to drive from City A to City B.
Choice C is correct. It’s given that d is the distance, in miles, Marisol traveled after driving for t hours. Therefore, 45 represents the distance in miles traveled per hour, which is the speed she drove in miles per hour.
Choice A is incorrect and may result from misidentifying speed as the number of trips. Choice B is incorrect and may result from misidentifying speed as the total distance. Choice D is incorrect and may result from misidentifying the speed as the time, in hours.
If , what is the value of ?
Choice C is correct. It’s given that . Substituting for into the given expression yields , which is equivalent to .
Choice A is incorrect. This is the value of .
Choice B is incorrect. This is the value of .
Choice D is incorrect. This is the value of .
For the linear function , and . Which equation defines ?
Choice C is correct. An equation defining the linear function can be written in the form , where is the slope and is the y-intercept of the graph of in the xy-plane. The slope of the graph of can be found using the formula , where and are any two points that the graph passes through. If , it follows that the graph of passes through the point . If , it follows that the graph of passes through the point . Substituting and for and , respectively, in the formula yields , which is equivalent to , or . Since the graph of passes through , it follows that . Substituting for and for in the equation yields , or . Thus, the equation that defines is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A gym charges its members a onetime enrollment fee and a membership fee of per month. If there are no charges other than the enrollment fee and the membership fee, after how many months will a member have been charged a total of at the gym?
Choice C is correct. It’s given that a gym charges its members a onetime enrollment fee and a membership fee of per month. Let represent the number of months at the gym after which a member will have been charged a total of . If there are no charges other than the enrollment fee and the membership fee, the equation can be used to represent this situation. Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, a member will have been charged a total of after months at the gym.
Choice A is incorrect. A member will have been charged a total of , or , after months at the gym.
Choice B is incorrect. A member will have been charged a total of , or , after months at the gym.
Choice D is incorrect. A member will have been charged a total of , or , after months at the gym.
If , what is the value of ?
Choice B is correct. Dividing both sides of the given equation by yields . Substituting for in the expression yields , which is equivalent to .
Choice A is incorrect. This is the value of .
Choice C is incorrect. This is the value of .
Choice D is incorrect. This is the value of .
The solution to the given system of equations is . What is the value of ?
Choice B is correct. Adding the first equation to the second equation in the given system yields , or . Combining like terms in this equation yields . Dividing both sides of this equation by yields . Thus, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of , not .
| Property address | Purchase price (dollars) | Monthly rental price (dollars) |
|---|---|---|
| Clearwater Lane | 128,000 | 950 |
| Driftwood Drive | 176,000 | 1,310 |
| Edgemont Street | 70,000 | 515 |
| Glenview Street | 140,000 | 1,040 |
| Hamilton Circle | 450,000 | 3,365 |
The Townsend Realty Group invested in the five different properties listed in the table above. The table shows the amount, in dollars, the company paid for each property and the corresponding monthly rental price, in dollars, the company charges for the property at each of the five locations. Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property?
$350,000
$291,700
$233,300
$175,000
Choice B is correct. Let x be the original price, in dollars, of the Glenview Street property. After the 40% discount, the price of the property became dollars, and after the additional 20% off the discounted price, the price of the property became
. Thus, in terms of the original price of the property, x, the purchase price of the property is
. It follows that
. Solving this equation for x gives
. Therefore, of the given choices, $291,700 best approximates the original price of the Glenview Street property.
Choice A is incorrect because it is the result of dividing the purchase price of the property by 0.4, as though the purchase price were 40% of the original price. Choice C is incorrect because it is the closest to dividing the purchase price of the property by 0.6, as though the purchase price were 60% of the original price. Choice D is incorrect because it is the result of dividing the purchase price of the property by 0.8, as though the purchase price were 80% of the original price.
If , what is the value of ?
Choice B is correct. Multiplying both sides of the given equation by yields , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A linear model estimates the population of a city from to . The model estimates the population was thousand in , thousand in , and thousand in . To the nearest whole number, what is the value of ?
The correct answer is . It’s given that a linear model estimates the population of a city from to . Since the population can be estimated using a linear model, it follows that there is a constant rate of change for the model. It’s also given that the model estimates the population was thousand in , thousand in , and thousand in . The change in the population between and is , or , thousand. The change in the number of years between and is , or , years. Dividing by gives , or , thousand per year. Thus, the change in population per year from to estimated by the model is thousand. The change in the number of years between and is , or , years. Multiplying the change in population per year by the change in number of years yields the increase in population from to estimated by the model: , or , thousand. Adding the change in population from to estimated by the model to the estimated population in yields the estimated population in . Thus, the estimated population in is , or , thousand. Therefore to the nearest whole number, the value of is .
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice A is correct. In each choice, the values of are , , and . Substituting the first value of , , for in the given inequality yields , or . Therefore, when , the corresponding value of must be greater than . Of the given choices, only choice A is a table where the value of corresponding to is greater than . To confirm that the other values of in this table and their corresponding values of are also solutions to the given inequality, the values of and in the table can be substituted for and in the given inequality. Substituting for and for in the given inequality yields , or , which is true. Substituting for and for in the given inequality yields , or , which is true. It follows that for choice A, all the values of and their corresponding values of are solutions to the given inequality.
Choice B is incorrect. Substituting for and for in the given inequality yields , or , which is false.
Choice C is incorrect. Substituting for and for in the given inequality yields , or , which is false.
Choice D is incorrect. Substituting for and for in the given inequality yields , or , which is false.
The table shows three values of and their corresponding values of . There is a linear relationship between and . Which of the following equations represents this relationship?
Choice D is correct. A linear relationship can be represented by an equation of the form , where and are constants. It’s given in the table that when , . Substituting for and for in yields , or . Substituting for in the equation yields . It’s also given in the table that when , . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Therefore, the equation represents the relationship between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . Solving by substitution, the given system of equations, where is a constant, can be written so that the left-hand side of each equation is equal to . Subtracting from each side of the first equation in the given system, , yields . Adding to each side of the second equation in the given system, , yields . Since the left-hand side of each equation is equal to , setting the the right-hand side of the equations equal to each other yields . A linear equation in one variable, , has no solution if and only if the equation is false; that is, when there's no value of that produces a true statement. For the equation , there's no value of that produces a true statement when . Therefore, for the equation , there's no value of that produces a true statement when the value of is . It follows that in the given system of equations, the system has no solution when the value of is .
On the first day of a semester, a film club has members. Each day after the first day of the semester, new members join the film club. If no members leave the film club, how many total members will the film club have days after the first day of the semester?
Choice B is correct. It’s given that the film club has members on the first day of a semester, and new members join the film club each day after the first day of the semester. This means that after days, , or , new members will have joined the club. Adding members to the original club members yields members. Thus, the film club will have total members days after the first day of the semester.
Choice A is incorrect. This is the number of members that will have joined the film club days after the first day of the semester if new members, not , join the film club each day.
Choice C is incorrect. This is the number of members the film club will have days after the first day of the semester if new member, not , joins the film club each day.
Choice D is incorrect. This is the number of members the film club has on the first day of the semester.
For what value of w does ?
Choice D is correct. To solve the equation, use the distributive property to multiply on the right-hand side of the equation which gives w – 10 = 2w + 10. Subtract w from both sides of the equation, which gives –10 = w + 10. Finally, subtract 10 from both sides of the equation, which gives –20 = w.
Choices A and B are incorrect and may result from making sign errors. Choice C is incorrect and may result from incompletely distributing the 2 in the expression 2(w + 5).
The shaded region shown represents solutions to an inequality. Which ordered pair is a solution to this inequality?
Choice D is correct. Since the shaded region shown represents solutions to an inequality, an ordered pair is a solution to the inequality if it's represented by a point in the shaded region. Of the given choices, only is represented by a point in the shaded region. Therefore, is a solution to the inequality.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
In the linear function , and . Which equation defines ?
Choice D is correct. Since is a linear function, it can be defined by an equation of the form , where and are constants. It's given that . Substituting for and for in the equation yields , or . Substituting for in the equation yields . It's given that . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Substituting for in the equation yields . Therefore, an equation that defines is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The given function represents the perimeter, in , of a rectangle with a length of and a fixed width. What is the width, in , of the rectangle?
Choice B is correct. It's given that represents the perimeter, in , of a rectangle with a length of and a fixed width. If represents a fixed width, in , then the perimeter, in , of a rectangle with a length of and a fixed width of can be given by the function . Therefore, . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the width, in , of the rectangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which of the following could be the second equation in the system?
Choice C is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Dividing both sides of the given equation by yields . Dividing both sides of choice C by also yields , so choice C is equivalent to the given equation. Thus, choice C could be the second equation in the system.
Choice A is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.
Choice B is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.
Choice D is incorrect. The system consisting of this equation and the given equation has no solution, not infinitely many solutions.
Line in the xy-plane is perpendicular to the line with equation
. What is the slope of line
?
0
The slope of line is undefined.
Choice A is correct. It is given that line is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is vertical, so
must therefore be horizontal. Horizontal lines have a slope of 0, so
has a slope of 0.
Choice B is incorrect. A line with slope is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has undefined slope (not slope of 2). Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope
. However, the line with equation x = 2 has undefined slope (not slope of
). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line perpendicular to it.
Which of the following points is the solution to the given system of equations in the xy-plane?
Choice B is correct. A solution to a system of equations in the xy-plane is a point that lies on the graph of each equation in the system. The first equation given is . Substituting for in the second given equation yields , or . It follows that in the xy-plane, the point lies on the graph of each equation in the system. Therefore, the solution to the given system of equations in the xy-plane is .
Choice A is incorrect. The point doesn't lie on the graph of either equation in the given system.
Choice C is incorrect. The point doesn't lie on the graph of the second equation in the given system.
Choice D is incorrect. The point doesn't lie on the graph of the second equation in the given system.
The function models the volume of liquid, in milliliters, in a container seconds after it begins draining from a hole at the bottom. According to the model, what is the predicted volume, in milliliters, draining from the container each second?
Choice D is correct. It’s given that the function models the volume of liquid, in milliliters, in a container seconds after it begins draining from a hole at the bottom. The given function can be rewritten as . Thus, for each increase of by , the value of decreases by , or . Therefore, the predicted volume, in milliliters, draining from the container each second is milliliters.
Choice A is incorrect. This is the amount of liquid, in milliliters, in the container before the liquid begins draining.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
Choice B is correct. It's given that line is defined by . For an equation in slope-intercept form , represents the slope of the line defined by this equation in the xy-plane and represents the y-coordinate of the y-intercept of this line. Therefore, the slope of line is . It’s also given that line is perpendicular to line in the xy-plane. Therefore, the slope of line is the opposite reciprocal of the slope of line . The opposite reciprocal of is . Therefore, the slope of line is .
Choice A is incorrect. This is the opposite reciprocal of the y-coordinate of the y-intercept, not the slope, of line .
Choice C is incorrect. This is the y-coordinate of the y-intercept of line , not the slope of line .
Choice D is incorrect. This is the slope of a line that is parallel, not perpendicular, to line .
Which equation has the same solution as the given equation?
Choice C is correct. It’s given that . Subtracting from each side of this equation yields . Therefore, the equation is equivalent to the given equation and has the same solution.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What value of p is the solution of the equation above?
The correct answer is 1.2. One way to solve the equation is to first distribute the terms outside the parentheses to the terms inside the parentheses:
. Next, combine like terms on the left side of the equal sign:
. Subtracting 10p from both sides yields
. Finally, dividing both sides by
gives
, which is equivalent to
. Note that 1.2 and 6/5 are examples of ways to enter a correct answer.
If , what is the value of ?
The correct answer is . Multiplying both sides of the given equation by yields , or . Therefore, the value of is .
The point is a solution to the system of inequalities in the xy-plane. Which of the following could be the value of ?
Choice A is correct. It’s given that the point is a solution to the given system of inequalities in the xy-plane. This means that the coordinates of the point, when substituted for the variables and , make both of the inequalities in the system true. Substituting for in the inequality yields , which is true. Substituting for in the inequality yields . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields . Therefore, must be a value less than . Of the given choices, only is less than .
Choice B is incorrect. Substituting for and for in the inequality yields , or , which is not true.
Choice C is incorrect. Substituting for and for in the inequality yields , or , which is not true.
Choice D is incorrect. Substituting for and for in the inequality yields , or , which is not true.
The perimeter of an isosceles triangle is feet. Each of the two congruent sides of the triangle has a length of feet. What is the length, in feet, of the third side?
Choice C is correct. It’s given that the perimeter of an isosceles triangle is feet and that each of the two congruent sides has a length of feet. The perimeter of a triangle is the sum of the lengths of its three sides. The equation can be used to represent this situation, where is the length, in feet, of the third side. Combining like terms on the left-hand side of this equation yields . Subtracting from each side of this equation yields . Therefore, the length, in feet, of the third side is .
Choice A is incorrect. This would be the length, in feet, of the third side if the perimeter was feet, not feet.
Choice B is incorrect. This would be the length, in feet, of the third side if the perimeter was feet, not feet.
Choice D is incorrect. This would be the length, in feet, of the third side if the perimeter was feet, not feet.
The solution to the given system of equations is . What is the value of ?
Choice B is correct. Adding the second equation in the given system to the first equation in the given system yields , which is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. If a temperature increased by kelvins, by how much did the temperature increase, in degrees Fahrenheit?
Choice A is correct. It’s given that the function gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of kelvins. A temperature that increased by kelvins means that the value of increased by kelvins. It follows that an increase in by increases by , or . Therefore, if a temperature increased by kelvins, the temperature increased by degrees Fahrenheit.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function defined by gives the estimated length, in inches, of a vine plant months after Tavon purchased it. Which of the following is the best interpretation of in this context?
Tavon will keep the vine plant for months.
The vine plant is expected to grow inches each month.
The vine plant is expected to grow to a maximum length of inches.
The estimated length of the vine plant was inches when Tavon purchased it.
Choice D is correct. It's given that the function defined by gives the estimated length, in inches, of a vine plant months after Tavon purchased it. For a function defined by an equation of the form , where and are constants, represents the value of , or the value of when the value of is . Therefore, for the function defined by , represents the value of when the value of is . This means that months after the vine plant was purchased, the estimated length of the vine plant was inches. Therefore, the best interpretation of in this context is the estimated length of the vine plant was inches when Tavon purchased it.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. The vine plant is expected to grow inches, not inches, each month.
Choice C is incorrect and may result from conceptual or calculation errors.
The equation gives the speed , in miles per hour, of a certain car seconds after it began to accelerate. What is the speed, in miles per hour, of the car seconds after it began to accelerate?
Choice D is correct. In the given equation, is the speed, in miles per hour, of a certain car seconds after it began to accelerate. Therefore, the speed of the car, in miles per hour, seconds after it began to accelerate can be found by substituting for in the given equation, which yields , or . Thus, the speed of the car seconds after it began to accelerate is miles per hour.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Each side of a -sided polygon has one of three lengths. The number of sides with length is times the number of sides with length . There are sides with length . Which equation must be true for the value of ?
Choice B is correct. It’s given that each side of a -sided polygon has one of three lengths. It's also given that the number of sides with length is times the number of sides with length . Therefore, there are , or , sides with length . It’s also given that there are sides with length . Therefore, the number of , , and sides are , , and , respectively. Since there are a total of sides, the equation represents this situation. Combining like terms on the left-hand side of this equation yields . Therefore, the equation that must be true for the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Connor has dollars and Maria has dollars. Connor has times as many dollars as Maria, and together they have a total of . Which system of equations represents this situation?
Choice A is correct. It’s given that Connor has dollars, Maria has dollars, and Connor has times as many dollars as Maria. This can be represented by the equation . It’s also given that together, Connor and Maria have a total of , which can be represented by the equation . Therefore, the system consisting of the equations and represents this situation.
Choice B is incorrect. The equation represents a situation where Maria has times as many dollars as Connor, rather than the situation where Connor has times as many dollars as Maria.
Choice C is incorrect. The equation represents a situation where Connor has times, rather than times, as many dollars as Maria. The equation represents a situation where Connor and Maria together have a total of , rather than .
Choice D is incorrect. The equation represents a situation where Maria has times as many dollars as Connor, rather than the situation where Connor has times as many dollars as Maria. The equation represents a situation where Connor and Maria together have a total of , rather than .
In the equation above, T represents Brittany’s total take-home pay, in dollars, for her first week of work, where h represents the number of hours she worked that week and 1,000 represents a sign-on bonus. If Brittany’s total take-home pay was $1,576, for how many hours was Brittany paid for her first week of work?
16
32
55
88
Choice B is correct. Since Brittany’s total take-home pay was $1,576, the value 1,576 can be substituted for T in the given equation to give
. Subtracting 1,000 from both sides of this equation gives
. Dividing both sides of this equation by 18 gives
. Therefore, Brittany was paid for 32 hours for her first week of work.
Choice A is incorrect. This is half the number of hours Brittany was paid for. Choice C is incorrect and may result from dividing 1,000 by 18. Choice D is incorrect and may result from dividing 1,576 by 18.
The function is defined as
, where a is a constant. If
, what is the value of a ?
Choice C is correct. Substituting 4 for x in g(x) = 5x + a gives g(4) = 5(4) + a. Since g(4) = 31, the equation g(4) = 5(4) + a simplifies to 31 = 20 + a. It follows that a = 11.
Choices A, B, and D are incorrect and may result from arithmetic errors.
For the linear function , is a constant and . What is the value of ?
The correct answer is . It’s given that for the linear function , is a constant and . Substituting for and for in the given equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the given equation, , yields . Substituting for in this equation yields , or . Therefore, the value of is .
Tony spends $80 per month on public transportation. A 10-ride pass costs $12.50, and a single-ride pass costs $1.50. If g represents the number of 10-ride passes Tony buys in a month and t represents the number of single-ride passes Tony buys in a month, which of the following equations best represents the relationship between g and t ?
Choice D is correct. Since a 10-ride pass costs $12.50 and g is the number of 10-ride passes Tony buys in a month, the expression represents the amount Tony spends on 10-ride passes in a month. Since a single-ride pass costs $1.50 and t is the number of single-ride passes Tony buys in a month, the expression
represents the amount Tony spends on single-ride passes in a month. Therefore, the sum
represents the amount he spends on the two types of passes in a month. Since Tony spends a total of $80 on passes in a month, this expression can be set equal to 80, producing
.
Choices A and B are incorrect. The expression represents the total number of the two types of passes Tony buys in a month, not the amount Tony spends, which is equal to 80, nor the cost of one of each pass, which is equal to
. Choice C is incorrect and may result from reversing the cost for each type of pass Tony buys in a month.
If , what is the value of ?
Choice C is correct. It’s given that . Multiplying each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
In an article about exercise, it is estimated that a 160-pound adult uses 200 calories for every 30 minutes of hiking and 150 calories for every 30 minutes of bicycling. An adult who weighs 160 pounds has completed 1 hour of bicycling. Based on the article, how many hours should the adult hike to use a total of 1,900 calories from bicycling and hiking?
9.5
8.75
6
4
Choice D is correct. Since a 160-pound adult uses 200 calories for every 30 minutes of hiking, then the same adult uses calories after hiking for h 30-minute periods. Similarly, the same adult uses
calories after bicycling for b 30-minute periods. Therefore, the equation
represents the situation where a 160-pound adult uses a total of 1,900 calories from hiking for h 30-minute periods and bicycling for b 30-minute periods. It’s given that the adult completes 1 hour, or 2 30-minute periods, of bicycling. Substituting 2 for b in the equation
yields
, or
. Subtracting 300 from both sides of this equation yields
. Dividing both sides by 200 yields
. Since h represents the number of 30-minute periods spent hiking and there are 2 30-minute periods in every hour, it follows that the adult will need to hike for
, or 4 hours to use a total of 1,900 calories from bicycling and hiking.
Choice A is incorrect and may result from solving the equation . This represents 0 30-minute periods bicycling instead of 2. Choice B is incorrect and may result from solving the equation
. This represents 1 30-minute period of bicycling instead of 2. Choice C is incorrect. This may result from determining that the number of 30-minute periods the adult should hike is 8, but then subtracting 2 from 8, rather than dividing 8 by 2, to find the number of hours the adult should hike.
In the linear function , and . Which equation defines ?
Choice D is correct. An equation defining can be written in the form , where , represents the slope of the graph of in the xy-plane, and represents the y-coordinate of the y-intercept of the graph. It’s given that and . It follows that the points and are on the graph of in the xy-plane. The slope can be found by using any two points, and , and the formula . Substituting and for and , respectively, in the slope formula yields , or . Substituting for and for in the equation yields , or . Adding to both sides of this equation yields . Substituting for and for in the equation yields . Since , it follows that the equation that defines is .
Choice A is incorrect. For this function, , not .
Choice B is incorrect. For this function, , not , and , not .
Choice C is incorrect. For this function, , not , and , not .
The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If x food calories is equivalent to k kilojoules, of the following, which best represents the relationship between x and k ?
Choice B is correct. The relationship between x food calories and k kilojoules can be modeled as a proportional relationship. Let and
represent the values in the first two rows in the table:
and
. The rate of change, or
, is
; therefore, the equation that best represents the relationship between x and k is
.
Choice A is incorrect and may be the result of calculating the rate of change using . Choice C is incorrect because the number of kilojoules is greater than the number of food calories. Choice D is incorrect and may be the result of an error when setting up the equation.
The solution to the given system of equations is . What is the value of ?
Choice A is correct. The first equation in the given system of equations is . Substituting for in the second equation in the given system of equations yields , or .
Choice B is incorrect. This is the value of in the solution to the given system of equations, not the value of .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which equation could be the second equation in this system?
Choice D is correct. It’s given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. When one equation is a multiple of another equation, the two equations are equivalent. Multiplying each side of the given equation by yields . Thus, is equivalent to the given equation and could be the second equation in the system.
Choice A is incorrect. The system consisting of this equation and the given equation has one solution rather than infinitely many solutions.
Choice B is incorrect. The system consisting of this equation and the given equation has one solution rather than infinitely many solutions.
Choice C is incorrect. The system consisting of this equation and the given equation has no solutions rather than infinitely many solutions.
The formula above is Ohm’s law for an electric circuit with current I, in amperes, potential difference V, in volts, and resistance R, in ohms. A circuit has a resistance of 500 ohms, and its potential difference will be generated by n six-volt batteries that produce a total potential difference of volts. If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, n, of six-volt batteries that can be used?
The correct answer is 20. For the given circuit, the resistance R is 500 ohms, and the total potential difference V generated by n batteries is volts. It’s also given that the circuit is to have a current of no more than 0.25 ampere, which can be expressed as
. Since Ohm’s law says that
, the given values for V and R can be substituted for I in this inequality, which yields
. Multiplying both sides of this inequality by 500 yields
, and dividing both sides of this inequality by 6 yields
. Since the number of batteries must be a whole number less than 20.833, the greatest number of batteries that can be used in this circuit is 20.
What value of is the solution to the given equation?
The correct answer is . Adding to both sides of the given equation yields . Dividing both sides of this equation by yields .
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice C is correct. All the tables in the choices have the same three values of , so each of the three values of can be substituted in the given inequality to compare the corresponding values of in each of the tables. Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . For the table in choice C, when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than . Therefore, the table in choice C gives values of and their corresponding values of that are all solutions to the given inequality.
Choice A is incorrect. In the table for choice A, when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than .
Choice B is incorrect. In the table for choice B, when , the corresponding value of is , which is not less than .
Choice D is incorrect. In the table for choice D, when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than .
If , what is the value of
?
2
Choice D is correct. Multiplying the given equation by 2 on each side yields . Applying the distributive property, this equation can be rewritten as
, or
.
Choices A, B, and C are incorrect and may result from calculation errors in solving the given equation for and then substituting that value of x in the expression
.
The table gives the number of hours, , of labor and a plumber’s total charge , in dollars, for two different jobs.
There is a linear relationship between and . Which equation represents this relationship?
Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, , and the plumber's total charge , in dollars. It follows that the relationship can be represented by an equation of the form , where is the rate of change of the function and is a constant. The rate of change of can be calculated by dividing the difference in two values of by the difference in the corresponding values of . Based on the values given in the table, the rate of change of is , or . Substituting for in the equation yields . The value of can be found by substituting any value of and its corresponding value of for and , respectively, in this equation. Substituting for and for yields , or . Subtracting from both sides of this equation yields . Substituting for in the equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For the linear function , the graph of in the xy-plane has a slope of and passes through the point . Which equation defines ?
Choice C is correct. An equation defining a linear function can be written in the form , where is the slope and is the y-intercept of the graph of in the xy-plane. It’s given that the graph of has a slope of . Therefore, . It’s also given that the graph of passes through the point . It follows that when , . Substituting for , for , and for in the equation yields , or . Subtracting from each side of this equation yields . Therefore, . Substituting for and for in the equation yields . Therefore, the equation that defines is .
Choice A is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
Choice B is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
Choice D is incorrect. For this function, the graph of in the xy-plane passes through the point , not .
A wire with a length of inches is cut into two parts. One part has a length of inches, and the other part has a length of inches. The value of is more than times the value of . What is the value of ?
Choice D is correct. It's given that a wire with a length of inches is cut into two parts. It's also given that one part has a length of inches and the other part has a length of inches. This can be represented by the equation . It's also given that the value of is more than times the value of . This can be represented by the equation . Substituting for in the equation yields , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Substituting for in the equation yields , or .
Choice A is incorrect. This value represents less than half of the total length of inches; however, represents the length of the longer part of the wire, since it's given that the value of is more than times the value of .
Choice B is incorrect. This value represents less than half of the total length of inches; however, represents the length of the longer part of the wire, since it's given that the value of is more than times the value of .
Choice C is incorrect. This represents a part that is more than the length of the other part, rather than more than times the length of the other part.
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice D is correct. All the tables in the choices have the same three values of , so each of the three values of can be substituted in the given inequality to compare the corresponding values of in each of the tables. Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is greater than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is greater than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is greater than . For the table in choice D, when , the corresponding value of is , which is greater than ; when , the corresponding value of is , which is greater than ; when , the corresponding value of is , which is greater than . Therefore, the table in choice D gives values of and their corresponding values of that are all solutions to the given inequality.
Choice A is incorrect. In the table for choice A, when , the corresponding value of is , which is not greater than ; when , the corresponding value of is , which is not greater than ; when , the corresponding value of is , which is not greater than .
Choice B is incorrect. In the table for choice B, when , the corresponding value of is , which is not greater than ; when , the corresponding value of is , which is not greater than .
Choice C is incorrect. In the table for choice C, when , the corresponding value of is , which is not greater than ; when , the corresponding value of is , which is not greater than ; when , the corresponding value of is , which is not greater than .
The solution to the given system of equations is . What is the value of ?
The correct answer is . Multiplying the first equation in the given system by yields . Subtracting the second equation in the given system, , from yields , which is equivalent to , or . Dividing both sides of this equation by yields . The value of can be found by substituting for in either of the two given equations. Substituting for in the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields , or . Therefore, the value of is , or . Note that 9/5 and 1.8 are examples of ways to enter a correct answer.
Normal body temperature for an adult is between and
, inclusive. If Kevin, an adult male, has a body temperature that is considered to be normal, which of the following could be his body temperature?
Choice C is correct. Normal body temperature must be greater than or equal to 97.8°F but less than or equal to 99°F. Of the given choices, 97.9°F is the only temperature that fits these restrictions.
Choices A and B are incorrect. These temperatures are less than 97.8°F, so they don’t fit the given restrictions. Choice D is incorrect. This temperature is greater than 99°F, so it doesn’t fit the given restrictions.
A total of squares each have side length . A total of equilateral triangles each have side length . None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is . Which equation represents this situation?
Choice C is correct. It’s given that a total of squares each have side length . Therefore, each of the squares has perimeter . Since there are a total of squares, the sum of the perimeters of these squares is , which is equivalent to , or . It’s also given that a total of equilateral triangles each have side length . Therefore, each of the equilateral triangles has perimeter . Since there are a total of equilateral triangles, the sum of the perimeters of these triangles is , which is equivalent to , or . Since the sum of the perimeters of the squares is and the sum of the perimeters of the triangles is , the sum of the perimeters of all these squares and triangles is . It’s given that the sum of the perimeters of all these squares and triangles is . Therefore, the equation represents this situation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph of is translated down units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?
The correct answer is . When the graph of an equation in the form , where , , and are constants, is translated down units in the xy-plane, the resulting graph can be represented by the equation . It’s given that the graph of is translated down units in the xy-plane. Therefore, the resulting graph can be represented by the equation , or . Adding to both sides of this equation yields . The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of in the equation when . Substituting for in the equation yields , or . Dividing both sides of this equation by yields . Therefore, the x-coordinate of the x-intercept of the resulting graph is . Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.
| Rock type | Weight per volume (lb/ft3) | Cost per pound |
|---|---|---|
| Basalt | 180 | $0.18 |
| Granite | 165 | $0.09 |
| Limestone | 120 | $0.03 |
| Sandstone | 135 | $0.22 |
A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. The equation gives the total cost, in dollars, of the rocks used in the project in terms of the number of ft3 of limestone, w, and the number of ft3 of basalt, z. All four rock types are used in the project. Which of the following is the best interpretation of 3,385.80 in this context?
The cost of the granite and sandstone needed for the project
The cost of the basalt and limestone needed for the project
The cost of the basalt needed for the project
The cost of the sandstone needed for the project
Choice A is correct. The table shows the cost of limestone is $0.03/lb, and the weight per volume for limestone is 120 lb/ft3. Therefore, the term represents the cost, in dollars, of w ft3 of limestone. Similarly, the term
represents the cost, in dollars, of z ft3 of basalt. The given equation shows that the total cost of all the rocks used in the project is $7,576.20. Since it’s given that all four rock types are used in the project, the remaining term, 3,385.80, represents the cost, in dollars, of the granite and sandstone needed for the project.
Choice B is incorrect. The cost of basalt and limestone needed for the project can be represented by . Choice C is incorrect. The cost of the basalt needed for the project can be represented by the expression
. Choice D is incorrect and may result from neglecting to include granite in the rock types used for the project.
In the given equation, is a constant. The equation has no solution. What is the value of ?
Choice B is correct. A linear equation in one variable has no solution if and only if the equation is false; that is, when there is no value of that produces a true statement. It's given that in the equation , is a constant and the equation has no solution for . Therefore, the value of the constant is one that results in a false equation. Factoring out the common factor of on the left-hand side of the given equation yields . Dividing both sides of this equation by yields . Dividing both sides of this equation by yields . This equation is false if and only if . Adding to both sides of yields . Dividing both sides of this equation by yields . It follows that the equation is false if and only if . Therefore, the given equation has no solution if and only if the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following ordered pairs satisfies the inequality
?
I only
II only
I and II only
I and III only
Choice C is correct. Substituting into the inequality gives
, or
, which is a true statement. Substituting
into the inequality gives
, or
, which is a true statement. Substituting
into the inequality gives
, or
, which is not a true statement. Therefore,
and
are the only ordered pairs shown that satisfy the given inequality.
Choice A is incorrect because the ordered pair also satisfies the inequality. Choice B is incorrect because the ordered pair
also satisfies the inequality. Choice D is incorrect because the ordered pair
does not satisfy the inequality.
The solution to the given system of equations is . What is the value of y?
Choice A is correct. The given system of linear equations can be solved by the elimination method. Multiplying each side of the second equation in the given system by yields , or . Subtracting this equation from the first equation in the given system yields , which is equivalent to , or .
Choice B is incorrect. This is the value of , not the value of .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The equation gives the perimeter of a rectangular rug that has length , in feet, and width , in feet. The width of the rug is feet. What is the length, in feet, of the rug?
The correct answer is . It's given that the equation gives the perimeter of a rectangular rug that has length , in feet, and width , in feet. It's also given that the width of the rug is feet. Substituting for in the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Since represents the length, in feet, of the rug, it follows that the length of the rug is feet.
A bank account was opened with an initial deposit. Over the next several months, regular deposits were made into this account, and there were no withdrawals made during this time. The graph of the function shown, where , estimates the account balance, in dollars, in this bank account months since the initial deposit. To the nearest whole dollar, what is the amount of the initial deposit estimated by the graph?
The correct answer is . For the graph shown, the x-axis represents the time since the initial deposit, in months, and the y-axis represents the bank account balance, in dollars. The amount of the initial deposit is estimated by the y-coordinate of the point on the graph that represents months since the initial deposit. Therefore, the amount of the initial deposit is estimated by the corresponding y-value for the point when . When , it is estimated that . Thus, the amount of the initial deposit estimated by the graph, to the nearest whole dollar, is .
When line is graphed in the xy-plane, it has an x-intercept of and a y-intercept of . What is the slope of line ?
Choice C is correct. It's given that when line is graphed in the xy-plane, it has an x-intercept of and a y-intercept of . The slope, , of a line can be found using any two points on the line, and , and the slope formula . Substituting the points and for and , respectively, in the slope formula yields , or . Therefore, the slope of line is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the slope of a line that has an x-intercept of and a y-intercept of .
Choice D is incorrect and may result from conceptual or calculation errors.
A dance teacher ordered outfits for students for a dance recital. Outfits for boys cost $26, and outfits for girls cost $35. The dance teacher ordered a total of 28 outfits and spent $881. If b represents the number of outfits the dance teacher ordered for boys and g represents the number of outfits the dance teacher ordered for girls, which of the following systems of equations can be solved to find b and g ?
Choice B is correct. Outfits for boys cost $26 each and the teacher ordered b outfits for boys, so the teacher spent 26b dollars on outfits for boys. Similarly, outfits for girls cost $35 each and the teacher ordered g outfits for girls, so the teacher spent 35g dollars on outfits for girls. Since the teacher spent a total of $881 on outfits for boys and girls, the equation 26b + 35g = 881 must be true. And since the teacher ordered a total of 28 outfits, the equation b + g = 28 must also be true.
Choice A is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits. Choice C is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits, as well as switching the cost of the outfits for boys with the cost of the outfits for girls. Choice D is incorrect and may result from switching the cost of the outfits for boys with the cost of the outfits for girls.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Subtracting the second equation from the first equation in the given system of equations yields , which is equivalent to, or . Dividing each side of this equation by yields .
The function is defined by , and , where is a constant. What is the value of ?
Choice C is correct. It's given that and , where is a constant. Therefore, for the given function , when , . Substituting for and for in the given function yields . Multiplying both sides of this equation by yields . Subtracting from both sides of this equation yields . Therefore, the value of is .
Choice A is incorrect. This is the value of if .
Choice B is incorrect. This is the value of if .
Choice D is incorrect. This is the value of if .
In the system of equations above, c is a constant. If the system has no solution, what is the value of c ?
The correct answer is . A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-intercepts. Each of the given linear equations is written in the slope-intercept form,
, where m is the slope and b is the y-coordinate of the y-intercept of the graph of the equation. For these two linear equations, the y-intercepts are
and
. Thus, if the system of equations has no solution, the slopes of the graphs of the two linear equations must be the same. The slope of the graph of the first linear equation is
. Therefore, for the system of equations to have no solution, the value of c must be
. Note that 1/2 and .5 are examples of ways to enter a correct answer.
If , what is the value of ?
The correct answer is . Subtracting from both sides of the given equation yields . Applying the distributive property to the left-hand side of this equation yields . Adding to both sides of this equation yields . Subtracting from both sides of this equation yields . Therefore, the value of is .
If , what is the value of ?
The correct answer is . Dividing both sides of the equation by yields . Substituting for in the expression yields , or .
If , the value of is between which of the following pairs of values?
and
and
and
and
Choice B is correct. Multiplying both sides of the given equation by , or , yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, if , then the value of is . It follows that of the given choices, the value of is between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is $15 and the cost of each pair of pants is $25. Geoff can spend at most $120 at the store. If Geoff buys s shirts and p pairs of pants, which of the following must be true?
Choice A is correct. Since the cost of each shirt is $15 and Geoff buys s shirts, the expression represents the amount Geoff spends on shirts. Since the cost of each pair of pants is $25 and Geoff buys p pairs of pants, the expression
represents the amount Geoff spends on pants. Therefore, the sum
represents the total amount Geoff spends at the store. Since Geoff can spend at most $120 at the store, the total amount he spends must be less than or equal to 120. Thus,
.
Choice B is incorrect. This represents the situation in which Geoff spends at least, rather than at most, $120 at the store. Choice C is incorrect and may result from reversing the cost of a shirt and that of a pair of paints. Choice D is incorrect and may result from both reversing the cost of a shirt and that of a pair of pants and from representing a situation in which Geoff spends at least, rather than at most, $120 at the store.
If , what is the value of
?
24
49
50
99
Choice B is correct. Multiplying both sides of the given equation by 5 yields . Substituting 50 for
in the expression
yields
.
Alternate approach: Dividing both sides of by 2 yields
. Evaluating the expression
for
yields
.
Choice A is incorrect and may result from finding the value of instead of
. Choice C is incorrect and may result from finding the value of
instead of
. Choice D is incorrect and may result from finding the value of
instead of
.
In the xy-plane, line passes through the points and . Which equation defines line ?
Choice D is correct. An equation defining a line in the xy-plane can be written in the form , where represents the slope and represents the y-intercept of the line. It’s given that line passes through the point ; therefore, . The slope, , of a line can be found using any two points on the line, and , and the slope formula . Substituting the points and for and , respectively, in the slope formula yields , or . Substituting for and for in the equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
| x | y |
| 3 | 7 |
| k | 11 |
| 12 | n |
The table above shows the coordinates of three points on a line in the xy-plane, where k and n are constants. If the slope of the line is 2, what is the value of ?
The correct answer is 30. The slope of a line can be found by using the slope formula, . It’s given that the slope of the line is 2; therefore,
. According to the table, the points
and
lie on the line. Substituting the coordinates of these points into the equation gives
. Multiplying both sides of this equation by
gives
, or
. Solving for k gives
. According to the table, the points
and
also lie on the line. Substituting the coordinates of these points into
gives
. Solving for n gives
. Therefore,
, or 30.
The graph of a system of linear equations is shown. What is the solution to the system?
Choice B is correct. A solution to a system of equations must be the solution to each equation in the system. It follows that if is a solution to the system, the point lies on the graph in the xy-plane of each equation in the system. The point that lies on each graph of the system of linear equations shown is their intersection point . Therefore, the solution to the system is .
Choice A is incorrect. The point lies on one, but not both, of the graphs of the linear equations shown.
Choice C is incorrect. The point lies on one, but not both, of the graphs of the linear equations shown.
Choice D is incorrect. The point lies on one, but not both, of the graphs of the linear equations shown.
The solution to the given system of equations is . What is the value of ?
Choice D is correct. Multiplying the second equation in the given system by yields . Adding this equation to the first equation in the system yields , which is equivalent to , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not the value of .
Choice B is incorrect. This is the value of , not the value of .
Choice C is incorrect. This the value of , or , not the value of .
What value of is the solution to the given equation?
The correct answer is . Subtracting from each side of the given equation yields . Therefore, the value of that's the solution to the given equation is . Note that .2 and 1/5 are examples of ways to enter a correct answer.
The function is defined by the equation . What is the value of when ?
Choice D is correct. Substituting for in the given equation yields , or . Therefore, the value of when is .
Choice A is incorrect. This is the value of when .
Choice B is incorrect. This is the value of when .
Choice C is incorrect. This is the value of when .
A line in the xy-plane has a slope of and passes through the point . Which equation represents this line?
Choice D is correct. The equation of a line in the xy-plane can be written as , where represents the slope of the line and represents the y-intercept of the line. It's given that the slope of the line is . It follows that . It's also given that the line passes through the point . It follows that . Substituting for and for in yields . Thus, the equation represents this line.
Choice A is incorrect. This equation represents a line with a slope of and a y-intercept of .
Choice B is incorrect. This equation represents a line with a slope of and a y-intercept of .
Choice C is incorrect. This equation represents a line with a slope of and a y-intercept of .
What is the solution to the given system of equations?
Choice B is correct. The second equation in the given system is . Substituting for in the first equation in the given system yields , or . Subtracting from both sides of this equation yields . Therefore, the solution to the given system of equations is .
Choice A is incorrect. This is the solution , not , to the given system of equations.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Vivian bought party hats and cupcakes for . Each package of party hats cost , and each cupcake cost . If Vivian bought packages of party hats, how many cupcakes did she buy?
The correct answer is . The number of cupcakes Vivian bought can be found by first finding the amount Vivian spent on cupcakes. The amount Vivian spent on cupcakes can be found by subtracting the amount Vivian spent on party hats from the total amount Vivian spent. The amount Vivian spent on party hats can be found by multiplying the cost per package of party hats by the number of packages of party hats, which yields , or . Subtracting the amount Vivian spent on party hats, , from the total amount Vivian spent, , yields , or . Since the amount Vivian spent on cupcakes was and each cupcake cost , it follows that Vivian bought cupcakes.
The graph shows the linear relationship between and . Which table gives three values of and their corresponding values of for this relationship?
Choice D is correct. It’s given that the graph shows the linear relationship between and . The given graph passes through the points , , and . It follows that when , the corresponding value of is , when , the corresponding value of is , and when , the corresponding value of is . Of the given choices, only the table in choice D gives these three values of and their corresponding values of for the relationship shown in the graph.
Choice A is incorrect. This table represents a relationship between and such that the graph passes through the points , , and .
Choice B is incorrect. This table represents a relationship between and such that the graph passes through the points , , and .
Choice C is incorrect. This table represents a linear relationship between and such that the graph passes through the points , , and .
The length of a rectangle is inches and the width is inches. The perimeter is at most inches. Which inequality represents this situation?
Choice A is correct. The perimeter of a rectangle is equal to the sum of times its length and times its width. It's given that the rectangle's length is inches and the width is inches. Therefore, the perimeter, in inches, is , or , which is equivalent to . It's given that the perimeter is at most inches; therefore, represents this situation.
Choice B is incorrect. This inequality represents a situation where the perimeter is at least, rather than at most, inches.
Choice C is incorrect. This inequality represents a situation where times the length, rather than the length, is inches.
Choice D is incorrect. This inequality represents a situation where times the length, rather than the length, is inches, and the perimeter is at least, rather than at most, inches.
A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?
Choice A is correct. If x is the width, in inches, of the box, then the length of the box is 2.5x inches. It follows that the perimeter of the base is , or 7x inches. The height of the box is given to be 60 inches. According to the restriction, the sum of the perimeter of the base and the height of the box should not exceed 130 inches. Algebraically, this can be represented by
, or
. Dividing both sides of the inequality by 7 gives
. Since x represents the width of the box, x must also be a positive number. Therefore, the inequality
represents all the allowable values of x that satisfy the given conditions.
Choices B, C, and D are incorrect and may result from calculation errors or misreading the given information.
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
Choice D is correct. It’s given that line is defined by . This equation can be written in slope-intercept form , where is the slope of line and is the y-coordinate of the y-intercept of line . Adding to both sides of yields . Subtracting from both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, the slope of line is . It’s given that line is perpendicular to line in the xy-plane. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to divided by the slope of the second line. Therefore, the slope of line is the negative reciprocal of the slope of line . The negative reciprocal of is , or . Therefore, the slope of line is .
Choice A is incorrect. This is the slope of a line in the xy-plane that is parallel, not perpendicular, to line .
Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of .
Choice C is incorrect. This is the negative, not the negative reciprocal, of .
|
The table above shows some pairs of x values and y values. Which of the following equations could represent the relationship between x and y ?
Choice A is correct. Each of the choices is a linear equation in the form y = mx + b, where m and b are constants. In this equation, m represents the change in y for each increase in x by 1. From the table, it can be determined that the value of y increases by 2 for each increase in x by 1. In other words, for the pairs of x and y in the given table, m = 2. The value of b can be found by substituting the values of x and y from any row of the table and substituting the value of m into the equation y = mx + b and then solving for b. For example, using x = 1, y = 5, and m = 2 yields 5 = 2(1) + b. Solving for b yields b = 3. Therefore, the equation y = 2x + 3 could represent the relationship between x and y in the given table.
Alternatively, if an equation represents the relationship between x and y, then when each pair of x and y from the table is substituted into the equation, the result will be a true statement. Of the equations given, the equation y = 2x + 3 in choice A is the only equation that results in a true statement when each of the pairs of x and y are substituted into the equation.
Choices B, C, and D are incorrect because when at least one pair of x and y from the table is substituted into the equations given in these choices, the result is a false statement. For example, when the pair x = 4 and y = 11 is substituted into the equation in choice B, the result is 11 = 3(4) – 2, or 11 = 10, which is false.
| Rock type | Weight per volume (lb/ft3) | Cost per pound |
|---|---|---|
| Basalt | 180 | $0.18 |
| Granite | 165 | $0.09 |
| Limestone | 120 | $0.03 |
| Sandstone | 135 | $0.22 |
A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. Only basalt, granite, and limestone will be used in the garden. The rocks in the garden will have a total weight of 1,000 pounds. If 330 pounds of granite is used, which of the following equations could show the relationship between the amounts, x and y, in ft3, for each of the other rock types used?
Choice C is correct. It’s given that the weight of the granite used in the garden is 330 pounds. The weight of the limestone used in the garden is a product of its weight per volume, in lb/ft3, and its volume, in ft3. Therefore, the weight of the limestone used in the garden can be represented by , where x is the volume, in ft3, of the limestone used. Similarly, the weight of the basalt used in the garden can be represented by
, where y is the volume, in ft3, of the basalt used. It’s given that the total weight of the rocks used in the garden will be 1,000 pounds. Thus, the sum of the weights of the three rock types used is 1,000 pounds, which can be represented by the equation
. Subtracting 330 from both sides of this equation yields
.
Choice A is incorrect. This equation uses the weight per volume of granite instead of limestone. Choice B is incorrect. This equation uses the weight per volume of granite instead of basalt, and doesn’t take into account the 330 pounds of granite that will be used in the garden. Choice D is incorrect. This equation doesn’t take into account the 330 pounds of granite that will be used in the garden.
The solution to the given system of equations is . What is the value of ?
The correct answer is . The first equation in the given system is . Substituting for in the second equation in the given system yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the value of is .
If and , what is the value of ?
Choice C is correct. The value of can be found by substituting for in the given equation , which yields , or . The value of can be found by substituting for in the given equation , which yields , or . The value of the expression can be found by substituting the corresponding values into the expression, which gives . This expression is equivalent to , or .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from calculating as , rather than
.
Choice D is incorrect and may result from conceptual or calculation errors.
If is the solution to the given system of equations, what is the value of y ?
10
15
60
65
Choice B is correct. Substituting 20 for in the second equation in the system yields
, or
. Subtracting 40 from both sides of this equation yields
. Dividing both sides of this equation by 3 yields
.
Choice A is incorrect. If , then
since
. However, substituting 10 for both x and y in the second equation yields
, which is a false statement. Choice C is incorrect. If
, then
since
. However, substituting these values for x and y in the second equation yields
, which is a false statement. Choice D is incorrect. If
, then
since
. However, substituting these values for x and y in the second equation yields
, which is a false statement.
In the system of equations above, a is a constant. If the system of equations has no solution, what is the value of a ?
0
1
2
Choice D is correct. A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-coordinates of the y-intercepts. Each of the given equations is written in the slope-intercept form of a linear equation, , where m is the slope and b is the y-coordinate of the y-intercept of the graph of the equation. For these two linear equations, the y-coordinates of the y-intercepts are different:
and
. Thus, if the system of equations has no solution, the slopes of the two linear equations must be the same. The slope of the first linear equation is 2. Therefore, for the system of equations to have no solution, the value of a must be 2.
Choices A, B, and C are incorrect and may result from conceptual and computational errors.
For each real number , which of the following points lies on the graph of each equation in the xy-plane for the given system?
Choice A is correct. Dividing both sides of the second equation in the given system by yields , which is the first equation in the given system. Therefore, the first and second equations represent the same line in the xy-plane. If the x- and y-coordinates of a point satisfy an equation, the point lies on the graph of the equation in the xy-plane. Choice A is a point with x-coordinate and y-coordinate . Substituting for and for in the equation yields . Applying the distributive property to the left-hand side of this equation yields . Combining like terms on the left-hand side of this equation yields , so the coordinates of the point satisfy both equations in the given system. Therefore, for each real number , the point lies on the graph of each equation in the xy-plane for the given system.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For which of the following tables are all the values of and their corresponding values of solutions to the given inequality?
Choice C is correct. All the tables in the choices have the same three values of , so each of the three values of can be substituted in the given inequality to compare the corresponding values of in each of the tables. Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . Substituting for in the given inequality yields , or . Therefore, when , the corresponding value of is less than . For the table in choice C, when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than ; when , the corresponding value of is , which is less than . Therefore, the table in choice C gives values of and their corresponding values of that are all solutions to the given inequality.
Choice A is incorrect. In the table for choice A, when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than .
Choice B is incorrect. In the table for choice B, when , the corresponding value of is , which is not less than .
Choice D is incorrect. In the table for choice D, when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than ; when , the corresponding value of is , which is not less than .
In the given pair of equations, and are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?
Choice B is correct. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, , where is the slope of the graph of the equation in the xy-plane and is the y-intercept. For the first equation, , subtracting from both sides gives , and dividing both sides of this equation by gives . Therefore, the slope of the graph of this equation is . For the second equation, , subtracting from both sides gives , and dividing both sides of this equation by gives . Therefore, the slope of the graph of this equation is . Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, , must be the negative reciprocal of the slope of the graph of the first equation, . The negative reciprocal of is , or . Therefore, , or . Similarly, rewriting the equations in choice B in slope-intercept form yields and . It follows that the slope of the graph of the first equation in choice B is and the slope of the graph of the second equation in choice B is . Since , is equal to , or . Since is the negative reciprocal of , the pair of equations in choice B represents a pair of perpendicular lines.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
On January 1, 2015, a city’s minimum hourly wage was $9.25. It will increase by $0.50 on the first day of the year for the next 5 years. Which of the following functions best models the minimum hourly wage, in dollars, x years after January 1, 2015, where ?
Choice C is correct. It’s given that the city’s minimum hourly wage will increase by $0.50 on the first day of the year for the 5 years after January 1, 2015. Therefore, the total increase, in dollars, in the minimum hourly wage x years after January 1, 2015, is represented by . Since the minimum hourly wage on January 1, 2015, was $9.25, it follows that the minimum hourly wage, in dollars, x years after January 1, 2015, is represented by
. Therefore, the function
best models this situation.
Choices A, B, and D are incorrect. In choice A, the function models a situation where the minimum hourly wage is $9.25 on January 1, 2015, but decreases by $0.50 on the first day of the year for the next 5 years. The functions in choices B and D both model a situation where the minimum hourly wage is increasing by $9.25 on the first day of the year for the 5 years after January 1, 2015.
If , what is the value of ?
The correct answer is . Subtracting from each side of the given equation yields . Multiplying each side of this equation by yields . Multiplying each side of this equation by yields . Therefore, the value of is .
A business owner plans to purchase the same model of chair for each of the employees. The total budget to spend on these chairs is , which includes a sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?
Choice B is correct. It’s given that a business owner plans to purchase chairs. If is the price per chair, the total price of purchasing chairs is . It’s also given that sales tax is included, which is equivalent to multiplied by , or . Since the total budget is , the inequality representing the situation is given by . Dividing both sides of this inequality by and rounding the result to two decimal places gives . To not exceed the budget, the maximum possible price per chair is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the maximum possible price per chair including sales tax, not the maximum possible price per chair before sales tax.
Choice D is incorrect. This is the maximum possible price if the sales tax is added to the total budget, not the maximum possible price per chair before sales tax.
The solution to the given system of equations is . What is the value of ?
Choice C is correct. The given system of linear equations can be solved by the substitution method. Substituting for from the first equation in the given system into the second equation yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the absolute value of , not the value of .
At a school fair, students can win colored tokens that are worth a different number of points depending on the color. One student won green tokens and red tokens worth a total of points. The given equation represents this situation. How many more points is a red token worth than a green token?
The correct answer is . It's given that , where is the number of green tokens and is the number of red tokens won by one student and these tokens are worth a total of points. Since the equation represents the situation where the student won points with green tokens and red tokens for a total of points, each term on the left-hand side of the equation represents the number of points won for one of the colors. Since the coefficient of in the given equation is , a green token must be worth points. Similarly, since the coefficient of in the given equation is , a red token must be worth points. Therefore, a red token is worth points, or points, more than a green token.
In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer is represented by . The product of and the fourth odd integer is at most less than the sum of the first and third odd integers. Which inequality represents this situation?
Choice A is correct. It’s given that the four odd integers are consecutive, ordered from least to greatest, and that the first odd integer is represented by . It follows that the second odd integer is represented by , the third odd integer is represented by , and the fourth odd integer is represented by . Therefore, the product of and the fourth odd integer is represented by , and less than the sum of the first and third odd integers is represented by . Since the product of and the fourth odd integer is at most less than the sum of the first and third odd integers, it follows that .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A student council group is selling school posters for a fundraiser. They use the function to determine their profit , in dollars, for selling school posters. In order to earn a profit of , how many school posters must they sell?
The correct answer is . It’s given that a student council group uses the function to determine their profit , in dollars, for selling school posters. Substituting for in the given function yields . Adding to each side of this equation yields . Dividing each side of this equation by yields . Therefore, in order to earn a profit of , they must sell school posters.
Argon is placed inside a container with a constant volume. The graph shows the estimated pressure , in , of the argon when its temperature is kelvins.
What is the estimated pressure of the argon, in , when the temperature is kelvins?
Choice B is correct. For the graph shown, the x-axis represents temperature, in kelvins, and the y-axis represents the estimated pressure, in . The estimated pressure of the argon when the temperature is kelvins can be found by locating the point on the graph where the value of is equal to . The graph passes through the point . This means that when the temperature is kelvins, the estimated pressure is .
Choice A is incorrect. This is the estimated pressure, in , of the argon when the temperature is kelvins, not kelvins.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the temperature, in kelvins, of the argon.
The length, , of a white whale was when it was born and increased an average of per month for the first months after it was born. Which equation best represents this situation, where is the number of months after the whale was born and is the length, in , of the whale?
Choice D is correct. It's given that the length of the whale was when it was born and that its length increased an average of per month for the first months after it was born. Since represents the number of months after the whale was born, the total increase in the whale's length, in , is times , or . The length of the whale , in , can be found by adding the whale's length at birth, , to the total increase in length, . Therefore, the equation that best represents this situation is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The shaded region shown represents the solutions to which inequality?
Choice A is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form , where is the slope and is the y-intercept of the line. For the graph shown, the boundary line passes through the points and . Given two points on a line, and , the slope of the line can be calculated using the equation . Substituting the points and for and , respectively, in this equation yields , which is equivalent to , or . Since the point represents the y-intercept, it follows that . Substituting for and for in the equation yields as the equation of the boundary line. Since the shaded region represents all the points on and above this boundary line, it follows that the shaded region shown represents the solutions to the inequality .
Choice B is incorrect. This inequality represents a region whose boundary line has a y-intercept of , not .
Choice C is incorrect. This inequality represents a region whose boundary line has a y-intercept of , not .
Choice D is incorrect. This inequality represents a region whose boundary line has a y-intercept of , not .
Store A sells raspberries for per pint and blackberries for per pint. Store B sells raspberries for per pint and blackberries for per pint. A certain purchase of raspberries and blackberries would cost at Store A or at Store B. How many pints of blackberries are in this purchase?
Choice C is correct. It’s given that store A sells raspberries for per pint and blackberries for per pint, and a certain purchase of raspberries and blackberries at store A would cost . It’s also given that store B sells raspberries for per pint and blackberries for per pint, and this purchase of raspberries and blackberries at store B would cost . Let represent the number of pints of raspberries and represent the number of pints of blackberries in this purchase. The equation represents this purchase of raspberries and blackberries from store A and the equation represents this purchase of raspberries and blackberries from store B. Solving the system of equations by elimination gives the value of and the value of that make the system of equations true. Multiplying both sides of the equation for store A by yields , or . Multiplying both sides of the equation for store B by yields , or . Subtracting both sides of the equation for store A, , from the corresponding sides of the equation for store B, , yields , or . Dividing both sides of this equationby yields . Thus, pints of blackberries are in
this purchase.
Choices A and B are incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the number of pints of raspberries, not blackberries, in the purchase.
The equation represents the number of minutes of daylight (between sunrise and sunset), , and the number of minutes of non-daylight, , on a particular day in Oak Park, Illinois. If this day has minutes of daylight, how many minutes of non-daylight does it have?
Choice B is correct. It’s given that the equation represents the number of minutes of daylight, , and the number of minutes of non-daylight, , on a particular day in Oak Park, Illinois. It’s also given that this day has minutes of daylight. Substituting for in the equation yields . Subtracting from both sides of this equation yields . Therefore, this day has minutes of non-daylight.
Choice A is incorrect. This is the number of minutes of daylight, not non-daylight, on this day.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the total number of minutes of daylight and non-daylight.
| x | ||||
| f(x) | 21 | 18 | 15 | 12 |
The table above shows some values of x and their corresponding values for the linear function f. What is the x-intercept of the graph of
in the xy-plane?
Choice B is correct. The equation of a linear function can be written in the form , where
, m is the slope of the graph of
, and b is the y-coordinate of the y-intercept of the graph. The value of m can be found using the slope formula,
. According to the table, the points
and
lie on the graph of
. Using these two points in the slope formula yields
, or
. Substituting
for m in the slope-intercept form of the equation yields
. The value of b can be found by substituting values from the table and solving; for example, substituting the coordinates of the point
into the equation
gives
, which yields
. This means the function given by the table can be represented by the equation
. The value of the x-intercept of the graph of
can be determined by finding the value of x when
. Substituting
into
yields
, or
. This corresponds to the point
.
Choice A is incorrect and may result from substituting the value of the slope for the x-coordinate of the x-intercept. Choice C is incorrect and may result from a calculation error. Choice D is incorrect and may result from substituting the y-coordinate of the y-intercept for the x-coordinate of the x-intercept.
If , what is the value of x ?
3
6
Choice C is correct. To make it easier to add like terms on the left-hand side of the given equation, both sides of the equation can be multiplied by 6, which is the lowest common denominator of and
. This yields
, which can be rewritten as
. Dividing both sides of this equation by 2 yields
.
Choice A is incorrect and may result from subtracting the denominators instead of numerators with common denominators to get , rather than
, on the left-hand side of the equation. Choice B is incorrect and may result from rewriting the given equation as
instead of
. Choice D is incorrect and may result from conceptual or computational errors.
What is the slope of the graph of in the xy-plane?
The correct answer is . The graph of a line in the xy-plane can be represented by the equation , where is the slope of the line and is the y-coordinate of the y-intercept. The given equation can be written as . Therefore, the slope of the graph of this equation in the xy-plane is . Note that 5/13, .3846, 0.385, and 0.384 are examples of ways to enter a correct answer.
Which of the following systems of linear equations has no solution?
Choice A is correct. A system of two linear equations in two variables, and , has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, , where and are constants, are parallel if their slopes, , are the same and are distinct if their y-coordinates of the y-intercepts, , are different. In the equations and , the values of are each , and the values of are and , respectively. Since the slopes of these lines are the same and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice A has no solution.
Choice B is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, , rather than no solution.
Choice C is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, , rather than no solution.
Choice D is incorrect. The two lines represented by these equations are a vertical line and a horizontal line. Therefore, this system has a solution, , rather than no solution.
The function is defined by . What is the y-intercept of the graph of in the xy-plane?
Choice B is correct. The y-intercept of the graph of a function in the xy-plane is the point on the graph where . It′s given that . Substituting for in this equation yields , or . Since it′s given that , it follows that when . Therefore, the y-intercept of the graph of in the xy-plane is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given function models the number of gallons of gasoline that remains from a full gas tank in a car after driving miles. According to the model, about how many gallons of gasoline are used to drive each mile?
Choice A is correct. It's given that the function models the number of gallons that remain from a full gas tank in a car after driving miles. In the given function , the coefficient of is . This means that for every increase in the value of by , the value of decreases by . It follows that for each mile driven, there is a decrease of gallons of gasoline. Therefore, gallons of gasoline are used to drive each mile.
Choice B is incorrect and represents the number of gallons of gasoline in a full gas tank.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Which of the following systems of linear equations has no solution?
Choice C is correct. A system of two linear equations in two variables, and , has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, , where and are constants, are parallel if their slopes, , are the same and are distinct if their y-coordinates of the y-intercepts, , are different. In the equations and , the values of are each , and the values of are and , respectively. Since the slopes of these lines are the same, and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice C has no solution.
Choice A is incorrect. The lines represented by the equations in this system are a vertical line and a horizontal line. Therefore, this system has a solution, , rather than no solution.
Choice B is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, , rather than no solution.
Choice D is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, , rather than no solution.
Line is defined by . Line is parallel to line in the xy-plane. What is the slope of ?
The correct answer is . It's given that line is defined by . It's also given that line is parallel to line in the xy-plane. A line in the xy-plane represented by an equation in slope-intercept form has a slope of and a y-intercept of . Therefore, the slope of line is . Since parallel lines have equal slopes, the slope of line is . Note that 1/4 and .25 are examples of ways to enter a correct answer.
What value of is the solution to the given equation?
Choice A is correct. Applying the distributive property on the left-hand side of the given equation yields , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the solution to the equation , not .
Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?
38
33
22
16
Choice C is correct. Ken earned $8 per hour for the first 10 hours he worked, so he earned a total of $80 for the first 10 hours he worked. For the rest of the week, Ken was paid at the rate of $10 per hour. Let x be the number of hours he will work for the rest of the week. The total of Ken’s earnings, in dollars, for the week will be . He saves 90% of his earnings each week, so this week he will save
dollars. The inequality
represents the condition that he will save at least $270 for the week. Factoring 10 out of the expression
gives
. The product of 10 and 0.9 is 9, so the inequality can be rewritten as
. Dividing both sides of this inequality by 9 yields
, so
. Therefore, the least number of hours Ken must work the rest of the week to save at least $270 for the week is 22.
Choices A and B are incorrect because Ken can save $270 by working fewer hours than 38 or 33 for the rest of the week. Choice D is incorrect. If Ken worked 16 hours for the rest of the week, his total earnings for the week will be , which is less than $270. Since he saves only 90% of his earnings each week, he would save even less than $240 for the week.
The graph of a system of linear equations is shown. The solution to the system is . What is the value of ?
The correct answer is . A solution to a system of equations must satisfy each equation in the system. It follows that if is a solution to the system, the point lies on the graph in the xy-plane of each equation in the system. According to the graph, the point that lies on the graph of each equation in the system is . Therefore, the solution to the system is . It follows that the value of is .
Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?
Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting 1 from each side of the inequality gives
. So the given system of inequalities is equivalent to the system of inequalities
and
, which can be rewritten as
. Using the transitive property of inequalities, it follows that
.
Choice A is incorrect because there are points with a y-coordinate less than 6 that satisfy the given system of inequalities. For example, satisfies both inequalities. Choice C is incorrect. This may result from solving the inequality
for x, then replacing x with y. Choice D is incorrect because this inequality allows y-values that are not the y-coordinate of any point that satisfies both inequalities. For example,
is contained in the set
; however, if 2 is substituted into the first inequality for y, the result is
. This cannot be true because the second inequality gives
.
Kaylani used fabric measuring yards in length to make each suit for a men's choir. The relationship between the number of suits that Kaylani made, , and the total length of fabric that she purchased , in yards, is represented by the equation . What is the best interpretation of in this context?
Kaylani made suits.
Kaylani purchased a total of yards of fabric.
Kaylani used a total of yards of fabric to make the suits.
Kaylani purchased yards more fabric than she used to make the suits.
Choice D is correct. It’s given that the equation represents the relationship between the number of suits that Kaylani made, , and the total length of fabric she purchased, , in yards. Adding to both sides of the given equation yields . Since Kaylani made suits and used yards of fabric to make each suit, the expression represents the total amount of fabric she used to make the suits. Since represents the total length of fabric Kaylani purchased, in yards, it follows from the equation that Kaylani purchased yards of fabric to make the suits, plus an additional yards of fabric. Therefore, the best interpretation of in this context is that Kaylani purchased yards more fabric than she used to make the suits.
Choice A is incorrect. Kaylani made a total of suits, not suits.
Choice B is incorrect. Kaylani purchased a total of yards of fabric, not a total of yards of fabric.
Choice C is incorrect. Kaylani used a total of yards of fabric to make the suits, not a total of yards of fabric.
For a camping trip a group bought one-liter bottles of water and three-liter bottles of water, for a total of liters of water. Which equation represents this situation?
Choice A is correct. It's given that for a camping trip a group bought one-liter bottles of water and three-liter bottles of water. Since the group bought one-liter bottles of water, the total number of liters bought from one-liter bottles of water is represented as , or . Since the group bought three-liter bottles of water, the total number of liters bought from three-liter bottles of water is represented as . It's given that the group bought a total of liters; thus, the equation represents this situation.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect. This equation represents a situation where the group bought three-liter bottles of water and one-liter bottles of water, for a total of liters of water.
To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of numbers of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?
128
160
200
320
Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn dollars. Avery earns $20 per hour working at job B. Therefore, if she works b hours at job B, she will earn
dollars. The graph shown represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked b hours at job B, and earned s dollars from both jobs, the following equation represents the graph:
, where s is a constant. Identifying any point
from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value of s. For example, the point
, where
and
, lies on the graph. Substituting 16 for a and 0 for b in the equation
yields
, or
. Similarly, the point
, where
and
, lies on the graph. Substituting 0 for a and 8 for b in the equation
yields
, or
.
Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points on the graph will satisfy this equation. For example, if the value of s is 128 (choice A), then the equation
becomes
. The point
, where
and
, lies on the graph. However, substituting 16 for a and 0 for b in
yields
, or
, which is false. Therefore,
doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if
(choice C) or
(choice D), then substituting 16 for a and 0 for b yields
and
, respectively, which are both false.
If , what is the value of ?
The correct answer is . Multiplying both sides of the given equation by yields , or . Therefore, if , the value of is .
The equation above represents the speed y, in feet per second, of Sheila’s bicycle x seconds after she applied the brakes at the end of a ride. If the equation is graphed in the xy-plane, which of the following is the best interpretation of the x-coordinate of the line’s x-intercept in the context of the problem?
The speed of Sheila’s bicycle, in feet per second, before Sheila applied the brakes
The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes
The number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop
The number of feet Sheila’s bicycle traveled from the time she began applying the brakes until the bicycle came to a complete stop
Choice C is correct. It’s given that for each point on the graph of the given equation, the x-coordinate represents the number of seconds after Sheila applied the brakes, and the y-coordinate represents the speed of Sheila’s bicycle at that moment in time. For the graph of the equation, the y-coordinate of the x-intercept is 0. Therefore, the x-coordinate of the x-intercept of the graph of the given equation represents the number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop.
Choice A is incorrect. The speed of Sheila’s bicycle before she applied the brakes is represented by the y-coordinate of the y-intercept of the graph of the given equation, not the x-coordinate of the x-intercept. Choice B is incorrect. The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes is represented by the slope of the graph of the given equation, not the x-coordinate of the x-intercept. Choice D is incorrect and may result from misinterpreting x as the distance, in feet, traveled after applying the brakes, rather than the time, in seconds, after applying the brakes.
To repair a refrigerator, a technician charges per hour for labor plus for parts. Which function f represents the total amount, in dollars, the technician will charge for this job if it takes hours?
Choice C is correct. It’s given that the technician charges per hour for labor. Therefore, if the job takes hours, the technician will charge , or , for labor. It’s also given that the technician charges for parts. Therefore, represents the total amount, in dollars, the technician will charge for this job if it takes hours.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This function represents the total amount, in dollars, the technician charges for labor only, not the total amount charged for labor and parts.
Choice D is incorrect. This function represents the total amount, in dollars, the technician would charge if the charge for parts were subtracted from, rather than added to, the charge for labor.
A teacher is creating an assignment worth points. The assignment will consist of questions worth point and questions worth points. Which equation represents this situation, where represents the number of -point questions and represents the number of -point questions?
Choice D is correct. Since represents the number of -point questions and represents the number of -point questions, the assignment is worth a total of , or , points. Since the assignment is worth points, the equation represents this situation.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Figure A and figure B are both regular polygons. The sum of the perimeter of figure A and the perimeter of figure B is inches. The equation represents this situation, where is the number of sides of figure A and is the number of sides of figure B. Which statement is the best interpretation of in this context?
Each side of figure B has a length of inches.
The number of sides of figure B is .
Each side of figure A has a length of inches.
The number of sides of figure A is .
Choice A is correct. It’s given that figure A and figure B (not shown) are both regular polygons and the sum of the perimeters of the two figures is . It’s also given that is the number of sides of figure A and is the number of sides of figure B, and that the equation represents this situation. Thus, and represent the perimeters, in inches, of figure A and figure B, respectively. Since represents the perimeter, in inches, of figure B and is the number of sides of figure B, it follows that each side of figure B has a length of .
Choice B is incorrect. The number of sides of figure B is , not .
Choice C is incorrect. Since the perimeter, in inches, of figure A is represented by , each side of figure A has a length of , not .
Choice D is incorrect. The number of sides of figure A is , not .
In the solution to the system of equations above, what is the value of y ?
9
15
Choice D is correct. Multiplying both sides of by 5 results in
. Multiplying both sides of
by 2 results in
. Subtracting the resulting equations yields
, which simplifies to
. Dividing both sides of
by
results in
.
Choices A and B are incorrect and may result from incorrectly subtracting the transformed equation. Choice C is incorrect and may result from finding the value of x instead of the value of y.
An elementary school teacher is ordering x workbooks and y sets of flash cards for a math class. The teacher must order at least 20 items, but the total cost of the order must not be over $80. If the workbooks cost $3 each and the flash cards cost $4 per set, which of the following systems of inequalities models this situation?
Choice A is correct. The total number of workbooks and sets of flash cards ordered is represented by x + y. Since the teacher must order at least 20 items, it must be true that x + y ≥ 20. Each workbook costs $3; therefore, 3x represents the cost, in dollars, of x workbooks. Each set of flashcards costs $4; therefore, 4y represents the cost, in dollars, of y sets of flashcards. It follows that the total cost for x workbooks and y sets of flashcards is 3x + 4y. Since the total cost of the order must not be over $80, it must also be true that 3x + 4y ≤ 80. Of the choices given, these inequalities are shown only in choice A.
Choice B is incorrect. The second inequality says that the total cost must be greater, not less than or equal to $80. Choice C incorrectly limits the cost by the minimum number of items and the number of items with the maximum cost. Choice D is incorrect. The first inequality incorrectly says that at most 20 items must be ordered, and the second inequality says that the total cost of the order must be at least, not at most, $80.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
The correct answer is . A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form , where , , and are constants, are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system, , can be written in standard form by adding to both sides of the equation, which yields , or . Multiplying each term in this equation by yields . The second equation in the given system, , can be written in standard form by subtracting and from both sides of the equation, which yields . Multiplying each term in this equation by yields . The coefficient of in the first equation, , is equal to the coefficient of in the second equation, . For the lines to be parallel, and for the coefficients for and in one equation to be proportional to the corresponding coefficients in the other equation, the coefficient of in the second equation must also be equal to the coefficient of in the first equation. Therefore, . Dividing both sides of this equation by yields , or . Therefore, if the given system of equations has no solution, the value of is . Note that 7/2 and 3.5 are examples of ways to enter a correct answer.
A shipment consists of -pound boxes and -pound boxes with a total weight of pounds. There are -pound boxes in the shipment. How many -pound boxes are in the shipment?
Choice D is correct. It's given that the shipment consists of -pound boxes and -pound boxes with a total weight of pounds. Let represent the number of -pound boxes and represent the number of -pound boxes in the shipment. Therefore, the equation represents this situation. It's given that there are -pound boxes in the shipment. Substituting for in the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Thus, there are -pound boxes in the shipment.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the number of -pound boxes in the shipment.
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
Choice D is correct. A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form , where , , and are constants, are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in the form by subtracting from both sides of the equation to yield . The second equation in the given system can be written in the form by subtracting from both sides of the equation to yield . The coefficient of in this second equation, , is times the coefficient of in the first equation, . For the lines to be parallel, the coefficient of in the second equation, , must also be times the coefficient of in the first equation, . Thus, , or . Therefore, if the given system has no solution, the value of is .
Choice A is incorrect. If the value of is , then the given system would have one solution, rather than no solution.
Choice B is incorrect. If the value of is , then the given system would have one solution, rather than no solution.
Choice C is incorrect. If the value of is , then the given system would have one solution, rather than no solution.
In the linear function , and . Which equation defines ?
Choice A is correct. An equation defining a linear function can be written in the form , where and are constants. It’s given that . Substituting for and for in the equation yields , or . Substituting for in the equation yields . It’s also given that . Substituting for and for in the equation yields , or . Subtracting from the left- and right-hand sides of this equation yields . Substituting for in the equation yields , or .
Choice B is incorrect. Substituting for and for in this equation yields , which isn't a true statement.
Choice C is incorrect. Substituting for and for in this equation yields , or , which isn't a true statement.
Choice D is incorrect. Substituting for in this equation yields , which isn't a true statement.
The given equation describes the relationship between the number of birds, , and the number of reptiles, , that can be cared for at a pet care business on a given day. If the business cares for reptiles on a given day, how many birds can it care for on this day?
Choice A is correct. The number of birds can be found by calculating the value of when in the given equation. Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, if the business cares for reptiles on a given day, it can care for birds on this day.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Hana deposited a fixed amount into her bank account each month. The function gives the amount, in dollars, in Hana's bank account after monthly deposits. What is the best interpretation of in this context?
With each monthly deposit, the amount in Hana's bank account increased by .
Before Hana made any monthly deposits, the amount in her bank account was .
After monthly deposit, the amount in Hana's bank account was .
Hana made a total of monthly deposits.
Choice A is correct. It's given that represents the number of monthly deposits. In the given function , the coefficient of is . This means that for every increase in the value of by , the value of increases by . It follows that with each monthly deposit, the amount in Hana's bank account increased by .
Choice B is incorrect. Before Hana made any monthly deposits, the amount in her bank account was .
Choice C is incorrect. After monthly deposit, the amount in Hana's bank account was .
Choice D is incorrect and may result from conceptual errors.
The graph of a system of two linear equations is shown. What is the solution to the system?
Choice B is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when and , or at the point . Therefore, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
How many solutions does the given equation have?
Zero
Exactly one
Exactly two
Infinitely many
Choice B is correct. Adding to each side of the given equation yields . Dividing each side of this equation by yields . This means that is the only solution to the given equation. Therefore, the given equation has exactly one solution.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
An event planner is planning a party. It costs the event planner a onetime fee of to rent the venue and per attendee. The event planner has a budget of . What is the greatest number of attendees possible without exceeding the budget?
The correct answer is . The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let be the number of attendees. The expression thus represents the total cost of the party. It's given that the budget is , so this situation can be represented by the inequality . The greatest number of attendees can be found by solving this inequality for . Subtracting from both sides of this inequality gives . Dividing both sides of this inequality by results in approximately . Since the question is stated in terms of attendees, rounding down to the nearest whole number, , gives the greatest number of attendees possible.
Megan’s regular wage at her job is p dollars per hour for the first 8 hours of work in a day plus 1.5 times her regular hourly wage for work in excess of 8 hours that day. On a given day, Megan worked for 10 hours, and her total earnings for that day were $137.50. What is Megan’s regular hourly wage?
$11.75
$12.50
$13.25
$13.75
Choice B is correct. Since p represents Megan’s regular pay per hour, 1.5p represents the pay per hour in excess of 8 hours. Since Megan worked for 10 hours, she must have been paid p dollars per hour for 8 of the hours plus 1.5p dollars per hour for the remaining 2 hours. Therefore, since Megan earned $137.50 for the 10 hours, the situation can be represented by the equation 137.5 = 8p + 2(1.5)p. Distributing the 2 in the equation gives 137.5 = 8p + 3p, and combining like terms gives 137.5 = 11p. Dividing both sides by 11 gives p = 12.5. Therefore, Megan’s regular wage is $12.50.
Choices A and C are incorrect and may be the result of calculation errors. Choice D is incorrect and may result from finding the average hourly wage that Megan earned for the 10 hours of work.
Which table gives three values of and their corresponding values of for the given equation?
Choice A is correct. Each of the given choices gives three values of : , , and . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Substituting for in the given equation yields , or . Therefore, when , the corresponding value of for the given equation is . Thus, if the three values of are , , and , then their corresponding values of are , , and , respectively, for the given equation.
Choice B is incorrect. This table gives three values of and their corresponding values of for the equation .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . For what value of does ?
Choice A is correct. It's given that the function is defined by . Substituting for in this equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the value of when is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of , not the value of when .
If , what is the value of
?
The correct answer is 24. Multiplying both sides of the given equation by 3 yields . Using the distributive property to rewrite the left-hand side of this equation yields
.
The graph of is shown.
Which equation defines the linear function ?
Choice B is correct. The graph of a line in the xy-plane can be represented by the equation , where is the slope of the line and is the y-intercept. The slope of a line that passes through the points and can be calculated using the formula . The line shown passes through the points and . Substituting and for and , respectively, in the formula yields , which is equivalent to , or . Since the line shown passes through the point , it follows that . Substituting for and for in the equation yields . It’s given that the graph shown is the graph of . Substituting for in the equation yields . Adding to both sides of this equation yields . Therefore, the equation defines the linear function .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A piece of wire with a length of inches is cut into two parts. One part has a length of inches, and the other part has a length of inches. The value of is more than times the value of . What is the value of ?
The correct answer is . It’s given that a piece of wire has a length of inches and is cut into two parts. It’s also given that one part has a length of inches and the other part has a length of inches. It follows that the equation represents this situation. It’s also given that the value of is more than times the value of , or . Substitutingfor in the equation yields . Combining like terms on the left-hand side of this equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields , or . Therefore, the value of is .
A bowl contains ounces of water. When the bowl is uncovered, the amount of water in the bowl decreases by ounce every days. If ounces of water remain in this bowl, for how many days has it been uncovered?
Choice D is correct. It’s given that the bowl starts with ounces of water and has ounces of water remaining after a period of time has passed. The amount of water the bowl has lost during the time period can be found by subtracting the remaining amount of water from the amount of water the bowl starts with, which yields ounces, or ounces. This means the bowl loses ounces of water during that period of time. It’s given that the amount of water decreases by ounce every days. Letting represent the number of days the bowl has been uncovered, it follows that . Multiplying both sides of this equation by yields . Therefore, the bowl has been uncovered for days.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of for the equation , not .
A model estimates that whales from the genus Eschrichtius travel to miles in the ocean each day during their migration. Based on this model, which inequality represents the estimated total number of miles, , a whale from the genus Eschrichtius could travel in days of its migration?
Choice B is correct. It's given that the model estimates that whales from the genus Eschrichtius travel to miles in the ocean each day during their migration. If one of these whales travels miles each day for days, then the whale travels miles total. If one of these whales travels miles each day for days, then the whale travels miles total. Therefore, the model estimates that in days of its migration, a whale from the genus Eschrichtius could travel at least and at most miles total. Thus, the inequality represents the estimated total number of miles, , a whale from the genus Eschrichtius could travel in days of its migration.
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
What is the solution to the given equation?
Choice B is correct. Subtracting from both sides of the given equation yields . Therefore, the solution to the given equation is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined as more than times a number . If is graphed in the -plane, what is the best interpretation of the -intercept?
When , the number is .
When the number is , .
The value of increases by for each increase of in the value of the number.
For each increase of in the value of the number, increases by .
Choice A is correct. It’s given that the function is defined as more than times a number . This can be represented by the equation . The x-intercept of the graph of in the xy-plane is the point where the graph intersects the x-axis, or the point on the graph where the value of is equal to . Substituting for in the equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, when , the number is .
Choice B is incorrect. This is the best interpretation of the y-intercept, not the x-intercept, of the graph of the function.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the best interpretation of the slope, not the x-intercept, of the graph of the function.
The line with the equation is graphed in the xy‑plane. What is the x-coordinate of the x‑intercept of the line?
The correct answer is 1.25. The y-coordinate of the x-intercept is 0, so 0 can be substituted for y, giving . This simplifies to
. Multiplying both sides of
by 5 gives
. Dividing both sides of
by 4 gives
, which is equivalent to 1.25. Note that 1.25 and 5/4 are examples of ways to enter a correct answer.
The function is defined by . What is the value of ?
Choice B is correct. Substituting for in the given function yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of .
Choice C is incorrect. This is the value of .
Choice D is incorrect. This is the value of .
What is the y-intercept of the graph of in the xy-plane?
Choice A is correct. In the xy-plane, the graph of an equation in the form , where and are constants, has a slope of and a y-intercept of . Therefore, the y-intercept of the graph of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
One of the two equations in a linear system is . The system has no solution. Which of the following could be the other equation in the system?
Choice B is correct. A system of two linear equations written in standard form has no solution when the equations are distinct and the ratio of the x-coefficient to the y-coefficient for one equation is equivalent to the ratio of the x-coefficient to the y-coefficient for the other equation. This ratio for the given equation is 2 to 6, or 1 to 3. Only choice B is an equation that isn’t equivalent to the given equation and whose ratio of the x-coefficient to the y-coefficient is 1 to 3.
Choice A is incorrect. Multiplying each of the terms in this equation by 2 yields an equation that is equivalent to the given equation. This system would have infinitely many solutions. Choices C and D are incorrect. The ratio of the x-coefficient to the y-coefficient in (choice C) is
to 2, or
to 1. This ratio in
(choice D) is 6 to 2, or 3 to 1. Since neither of these ratios is equivalent to that for the given equation, these systems would have exactly one solution.
The point with coordinates lies on the line shown. What is the value of ?
Choice C is correct. It's given from the graph that the points and lie on the line. For two points on a line, and , the slope of the line can be calculated using the slope formula . Substituting for and for in this formula, the slope of the line can be calculated as , or . It's also given that the point lies on the line. Substituting for , for , and for in the slope formula yields , or . Multiplying both sides of this equation by yields . Expanding the left-hand side of this equation yields . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields . Thus, the value of is .
Choice A is incorrect. This is the value of when .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
The correct answer is . Subtracting from each side of the given equation yields . Adding to each side of this equation yields . Therefore, the value of is .
How many solutions does the given equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice C is correct. If the two sides of a linear equation are equivalent, then the equation is true for any value. If an equation is true for any value, it has infinitely many solutions. Since the two sides of the given linear equation are equivalent, the given equation has infinitely many solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The solution to the given system of equations is . What is the value of ?
The correct answer is . The given system of equations can be solved by the substitution method. The first equation in the given system of equations is . Substituting for in the second equation in the given system yields . Adding to both sides of this equation yields . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields .
In the given equation, is a constant. If the equation has infinitely many solutions, what is the value of ?
Choice C is correct. If an equation has infinitely many solutions, then the two sides of the equation must be equivalent. Multiplying each side of the given equation by yields . Since is a common factor of both terms on the left-hand side of this equation, the equation can be rewritten as . The two sides of this equation are equivalent when . Therefore, if the given equation has infinitely many solutions, the value of is .
Alternate approach: If the given equation, , has infinitely many solutions, then both sides of this equation are equal for any value of . If , then substituting for in the given equation yields , or . Dividing both sides of this equation by yields .
Choice A is incorrect. If the value of is , the given equation is equivalent to , which has one solution, not infinitely many solutions.
Choice B is incorrect. If the value of is , the given equation is equivalent to , or , which has one solution, not infinitely many solutions.
Choice D is incorrect. If the value of is , the given equation is equivalent to , or , which has one solution, not infinitely many solutions.
The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If the 180 food calories in a granola bar come entirely from p grams of protein, f grams of fat, and c grams of carbohydrate, which of the following expresses f in terms of p and c ?
Choice B is correct. It is given that there are 4.0 food calories per gram of protein, 9.0 food calories per gram of fat, and 4.0 food calories per gram of carbohydrate. If 180 food calories in a granola bar came from p grams of protein, f grams of fat, and c grams of carbohydrate, then the situation can be represented by the equation . The equation can then be rewritten in terms of f by subtracting 4p and 4c from both sides of the equation and then dividing both sides of the equation by 9. The result is the equation
.
Choices A, C, and D are incorrect and may be the result of not representing the situation with the correct equation or incorrectly rewriting the equation in terms of f.
Which of the following systems of equations has the same solution as the system of equations graphed above?
Choice A is correct. The solution to a system of equations is the coordinates of the intersection point of the graphs of the equations in the xy-plane. Based on the graph, the solution to the given system of equations is best approximated as . In the xy-plane, the graph of
is a horizontal line on which every y-coordinate is 0, and the graph of
is a vertical line on which every x-coordinate is
. These graphs intersect at the point
. Therefore, the system of equations in choice A has the same solution as the given system.
Choices B, C, and D are incorrect. If graphed in the xy-plane, these choices would intersect at the points ,
, and
, respectively, not
.
A moving truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than pounds. What is the maximum number of boxes this truck can tow in a trailer with a weight of pounds if each box weighs pounds?
Choice A is correct. It’s given that the truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than pounds. If the trailer has a weight of pounds and each box weighs pounds, the expression , where is the number of boxes, gives the combined weight of the trailer and the boxes. Since the combined weight must be no more than pounds, the possible numbers of boxes the truck can tow are given by the inequality . Subtracting from both sides of this inequality yields . Dividing both sides of this inequality by yields , or is less than or equal to approximately . Since the number of boxes, , must be a whole number, the maximum number of boxes the truck can tow is the greatest whole number less than , which is .
Choice B is incorrect. Towing the trailer and boxes would yield a combined weight of pounds, which is greater than pounds.
Choice C is incorrect. Towing the trailer and boxes would yield a combined weight of pounds, which is greater than pounds.
Choice D is incorrect. Towing the trailer and boxes would yield a combined weight of pounds, which is greater than pounds.
An object hangs from a spring. The formula relates the length
, in centimeters, of the spring to the weight w, in newtons, of the object. Which of the following describes the meaning of the 2 in this context?
The length, in centimeters, of the spring with no weight attached
The weight, in newtons, of an object that will stretch the spring 30 centimeters
The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring
The increase in the length, in centimeters, of the spring for each one-newton increase in the weight of the object
Choice D is correct. The value 2 is multiplied by w, the weight of the object. When the weight is 0, the length is 30 + 2(0) = 30 centimeters. If the weight increases by w newtons, the length increases by 2w centimeters, or 2 centimeters for each one-newton increase in weight.
Choice A is incorrect because this describes the value 30. Choice B is incorrect because 30 represents the length of the spring before it has been stretched. Choice C is incorrect because this describes the value w.
The equation , where a and b are constants, has no solutions. Which of the following must be true?
I.
II.
III.
None
I only
I and II only
I and III only
Choice D is correct. For a linear equation in a form to have no solutions, the x-terms must have equal coefficients and the remaining terms must not be equal. Expanding the right-hand side of the given equation yields
. Inspecting the x-terms, 9 must equal a, so statement I must be true. Inspecting the remaining terms, 5 can’t equal
. Dividing both of these quantities by 9 yields that b can’t equal
. Therefore, statement III must be true. Since b can have any value other than
, statement II may or may not be true.
Choice A is incorrect. For the given equation to have no solution, both and
must be true. Choice B is incorrect because it must also be true that
. Choice C is incorrect because when
, there are many values of b that lead to an equation having no solution. That is, b might be 5, but b isn’t required to be 5.
What is the y-intercept of the line graphed?
Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The line graphed intersects the y-axis at the point . Therefore, the y-intercept of the line graphed is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
If is the solution to the given system of equations, what is the value of y ?
The correct answer is 23. Since it’s given that , the value of y can be found by substituting 2 for x in the first equation and solving for y. Substituting 2 for x yields
, or
. Subtracting 6 from both sides of this equation yields
.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Adding the first equation to the second equation in the given system yields , or . Multiplying both sides of this equation by yields . Therefore, the value of is .
In the equation above, F represents the total amount of money, in dollars, a food truck charges for x drinks and y salads. The price, in dollars, of each drink is the same, and the price, in dollars, of each salad is the same. Which of the following is the best interpretation for the number 7.00 in this context?
The price, in dollars, of one drink
The price, in dollars, of one salad
The number of drinks bought during the day
The number of salads bought during the day
Choice B is correct. It’s given that is equal to the total amount of money, in dollars, a food truck charges for x drinks and y salads. Since each salad has the same price, it follows that the total charge for y salads is
dollars. When
, the value of the expression
is
, or 7.00. Therefore, the price for one salad is 7.00 dollars.
Choice A is incorrect. Since each drink has the same price, it follows that the total charge for x drinks is dollars. Therefore, the price, in dollars, for one drink is 2.50, not 7.00. Choices C and D are incorrect. In the given equation, F represents the total charge, in dollars, when x drinks and y salads are bought at the food truck. No information is provided about the number of drinks or the number of salads that are bought during the day. Therefore, 7.00 doesn’t represent either of these quantities.
For the function f, if for all values of x, what is the value of
?
0
2
Choice B is correct. It’s given that for all values of x. If
, then
will equal
. Dividing both sides of
by 3 gives
. Therefore, substituting 2 for x in the given equation yields
, which can be rewritten as
.
Choice A is incorrect. This is the value of the constant in the given equation for f. Choice C is incorrect and may result from substituting , rather than
, into the given equation. Choice D is incorrect. This is the value of x that yields
for the left-hand side of the given equation; it’s not the value of
.
The solution to the given system of equations is . What is the value of ?
Choice D is correct. Adding the second equation to the first equation in the given system of equations yields , or . Adding to each side of this equation yields . Multiplying each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
A proposal for a new library was included on an election ballot. A radio show stated that times as many people voted in favor of the proposal as people who voted against it. A social media post reported that more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?
Choice A is correct. It's given that a radio show stated that times as many people voted in favor of the proposal as people who voted against it. Let represent the number of people who voted against the proposal. It follows that is the number of people who voted in favor of the proposal and , or , is how many more people voted in favor of the proposal than voted against it. It's also given that a social media post reported that more people voted in favor of the proposal than voted against it. Thus, . Since , the value of must be half of , or . Therefore, people voted against the proposal.
Choice B is incorrect. This is how many more people voted in favor of the proposal than voted against it, not the number of people who voted against the proposal.
Choice C is incorrect. This is the number of people who voted in favor of the proposal, not the number of people who voted against the proposal.
Choice D is incorrect and may result from conceptual or calculation errors.
A rocket contained kilograms (kg) of propellant before launch. Exactly seconds after launch, kg of this propellant remained. On average, approximately how much propellant, in kg, did the rocket burn each second after launch?
Choice A is correct. It’s given that the rocket contained of propellant before launch and had remaining exactly seconds after launch. Finding the difference between the amount, in , of propellant before launch and the remaining amount, in , of propellant after launch gives the amount, in , of propellant burned during the seconds: . Dividing the amount of propellant burned by the number of seconds yields . Thus, an average of of propellant burned each second after launch.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from finding the amount of propellant burned, rather than the amount of propellant burned each second.
The linear function is defined by , where is a constant. If , where is a constant, which of the following expressions represents the value of ?
Choice C is correct. It’s given that . Therefore, for the given linear function , when , . Substituting for and for in yields . Applying the distributive property to the right-hand side of this equation yields . Adding to both sides of this equation yields . Adding to both sides of this equation yields , or . Therefore, the expression that represents the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For each real number , which of the following points lies on the graph of each equation in the xy-plane for the given system?
Choice B is correct. The two given equations are equivalent because the second equation can be obtained from the first equation by multiplying each side of the equation by . Thus, the graphs of the equations are coincident, so if a point lies on the graph of one of the equations, it also lies on the graph of the other equation. A point lies on the graph of an equation in the xy-plane if and only if this point represents a solution to the equation. It is sufficient, therefore, to find the point that represents a solution to the first given equation. Substituting the x- and y-coordinates of choice B, and , for and , respectively, in the first equation yields , which is equivalent to , or . Therefore, the point represents a solution to the first equation and thus lies on the graph of each equation in the xy-plane for the given system.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . For what value of does ?
The correct answer is . Substituting for in the given function yields . Multiplying each side of this equation by yields , or . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Therefore, when the value of is .
In the xy-plane, line k is defined by . Line j is perpendicular to line k, and the y-intercept of line j is
. Which of the following is an equation of line j ?
Choice D is correct. It’s given that line j is perpendicular to line k and that line k is defined by the equation . This equation can be rewritten in slope-intercept form,
, where m represents the slope of the line and b represents the y-coordinate of the y-intercept of the line, by subtracting x from both sides of the equation, which yields
. Thus, the slope of line k is
. Since line j and line k are perpendicular, their slopes are opposite reciprocals of each other. Thus, the slope of line j is 1. It’s given that the y-intercept of line j is
. Therefore, the equation for line j in slope-intercept form is
, which can be rewritten as
.
Choices A, B, and C are incorrect and may result from conceptual or calculation errors.
The graph of a system of linear equations is shown. What is the solution to the system?
Choice A is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when and , or at the point . Therefore, the solution to the system is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A line in the xy-plane has a slope of and passes through the point . Which equation represents this line?
Choice B is correct. A line in the xy-plane with a slope of and a y-intercept of can be represented by the equation . It's given that the line has a slope of . Therefore, . It's also given that the line passes through the point . Therefore, . Substituting for and for in the equation yields . Therefore, the equation represents this line.
Choice A is incorrect. This equation represents a line in the xy-plane that passes through the point , not .
Choice C is incorrect. This equation represents a line in the xy-plane that has a slope of , not , and passes through the point , not .
Choice D is incorrect. This equation represents a line in the xy-plane that has a slope of , not .
The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?
Choice B is correct. The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is dollars. In t years, the savings will be
dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is
.
Choice A is incorrect. It gives the number of years after installation at which the total amount of energy cost savings will be less than the installation cost. Choice C is incorrect and may result from subtracting the average annual energy cost for the home from the onetime cost
of the geothermal heating system installation. To find the predicted total savings, the predicted average cost should be subtracted from the average annual energy cost before the installation, and the result should be multiplied by the number of years, t. Choice D is incorrect and may result from misunderstanding the context. The ratio compares the average energy cost before installation and the average energy cost after installation; it does not represent the savings.
Henry receives a gift card to pay for movies online. He uses his gift card to buy movies for each. If he spends the rest of his gift card balance on renting movies for each, how many movies can Henry rent?
Choice B is correct. It's given that Henry uses his gift card to buy movies for each. Therefore, Henry spends , or , of his gift card to buy movies. After buying movies with his gift card, Henry has a gift card balance of , or . It's also given that Henry spends the rest of his gift card balance on renting movies for each. Therefore, Henry can rent , or , movies.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The table shows three values of and their corresponding values of , where is a constant, for the linear relationship between and . What is the slope of the line that represents this relationship in the xy-plane?
Choice A is correct. The slope, , of a line in the xy-plane can be found using two points on the line, and , and the slope formula . Based on the given table, the line representing the relationship between and in the xy-plane passes through the points , , and , where is a constant. Substituting two of these points, and , for and , respectively, in the slope formula yields , which is equivalent to , or . Therefore, the slope of the line that represents this relationship in the xy-plane is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The minimum value of is less than times another number . Which inequality shows the possible values of ?
Choice B is correct. It’s given that the minimum value of is less than times another number . Therefore, the possible values of are all greater than or equal to the value of less than times . The value of times is given by the expression . The value of less than is given by the expression . Therefore, the possible values of are all greater than or equal to . This can be shown by the inequality .
Choice A is incorrect. This inequality shows the possible values of if the maximum, not the minimum, value of is less than times .
Choice C is incorrect. This inequality shows the possible values of if the maximum, not the minimum, value of is times less than , not less than times .
Choice D is incorrect. This inequality shows the possible values of if the minimum value of is times less than , not less than times .
The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w ?
Choice B is correct. It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the dance floor is . So the perimeter is
.
Choice A is incorrect because it is the sum of one length and one width, which is only half the perimeter. Choice C is incorrect and may result from using the formula for the area instead of the formula for the perimeter and making a calculation error. Choice D is incorrect because this is the area, not the perimeter, of the dance floor.
A neighborhood consists of a -hectare park and a -hectare residential area. The total number of trees in the neighborhood is . The equation represents this situation. Which of the following is the best interpretation of x in this context?
The average number of trees per hectare in the park
The average number of trees per hectare in the residential area
The total number of trees in the park
The total number of trees in the residential area
Choice A is correct. It's given that a neighborhood consists of a -hectare park and a -hectare residential area and that the total number of trees in the neighborhood is . It's also given that the equation represents this situation. Since the total number of trees for a given area can be determined by taking the number of hectares times the average number of trees per hectare, this must mean that the terms and correspond to the number of trees in the park and in the residential area, respectively. Since corresponds to the number of trees in the park, and is the size of the park, in hectares, must represent the average number of trees per hectare in the park.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Jay walks at a speed of miles per hour and runs at a speed of miles per hour. He walks for hours and runs for hours for a combined total of miles. Which equation represents this situation?
Choice A is correct. Since Jay walks at a speed of miles per hour for hours, Jay walks a total of miles. Since Jay runs at a speed of miles per hour for hours, Jay runs a total of miles. Therefore, the total number of miles Jay travels can be represented by . Since the combined total number of miles is , the equation represents this situation.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Scientists collected fallen acorns that each housed a colony of the ant species P. ohioensis and analyzed each colony's structure. For any of these colonies, if the colony has worker ants, the equation , where , gives the predicted number of larvae, , in the colony. If one of these colonies has worker ants, which of the following is closest to the predicted number of larvae in the colony?
Choice A is correct. It's given that the equation , where , gives the predicted number of larvae, , in a colony of ants if the colony has worker ants. If one of these colonies has worker ants, the predicted number of larvae in that colony can be found by substituting for in the given equation. Substituting for in the given equation yields , or . Of the given choices, is closest to the predicted number of larvae in the colony.
Choice B is incorrect. This is closest to the predicted number of larvae in a colony with worker ants.
Choice C is incorrect. This is closest to the number of worker ants for which the predicted number of larvae in a colony is .
Choice D is incorrect. This is closest to the predicted number of larvae in a colony with worker ants.
A librarian has 43 books to distribute to a group of children. If he gives each child 2 books, he will have 7 books left over. How many children are in the group?
15
18
25
29
Choice B is correct. Subtracting the number of books left over from the total number of books results in , which is the number of books distributed. Dividing the number of books distributed by the number of books given to each child results in
.
Choice A is incorrect and results from dividing the total number of books by the number of books given to each child, , then subtracting the number of books left over from the result,
. Choice C is incorrect and results from adding the number of books left over to the total number of books,
, then dividing the result by the number of books given to each child,
. Choice D is incorrect and results from dividing the total number of books by the number of books given to each child,
, then adding the number of books left over,
.
A manager is responsible for ordering supplies for a shaved ice shop. The shop's inventory starts with paper cups, and the manager estimates that of these paper cups are used each day. Based on this estimate, in how many days will the supply of paper cups reach ?
Choice B is correct. It’s given that the shop’s inventory starts with paper cups and that the manager estimates that of these paper cups are used each day. Let represent the number of days in which the estimated supply of paper cups will reach . The equation represents this situation. Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, based on this estimate, the supply of paper cups will reach in days.
Choice A is incorrect. After days, the estimated supply of paper cups would be , or cups, not cups.
Choice C is incorrect. After days, the estimated supply of paper cups would be , or cups, not cups.
Choice D is incorrect. After days, the estimated supply of paper cups would be , or cups, which isn't possible.
The graph of the function f is a line in the xy-plane. If the line has slope and
, which of the following defines f?
Choice B is correct. The equation for the function f in the xy-plane can be represented by , where m is the slope and b is the y-coordinate of the y-intercept. Since it’s given that the line has a slope of
, it follows that
in
, which yields
. It’s given that
. This implies that the graph of the function f in the xy-plane passes through the point
. Thus, the y-coordinate of the y-intercept of the graph is 3, so
in
, which yields
. Therefore, the equation
defines the function f.
Choice A is incorrect and may result from a sign error for the y-intercept. Choice C is incorrect and may result from using the denominator of the given slope as m in , in addition to a sign error for the y-intercept. Choice D is incorrect and may result from using the denominator of the given slope as m in
.
The solution to the given system of equations is . What is the value of ?
Choice B is correct. The second equation in the given system is . Substituting for in the first equation in the given system yields , which is equivalent to . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph represents the total charge, in dollars, by an electrician for hours of work. The electrician charges a onetime fee plus an hourly rate. What is the best interpretation of the slope of the graph?
The electrician’s hourly rate
The electrician’s onetime fee
The maximum amount that the electrician charges
The total amount that the electrician charges
Choice A is correct. It’s given that the electrician charges a onetime fee plus an hourly rate. It's also given that the graph represents the total charge, in dollars, for hours of work. This graph shows a linear relationship in the xy-plane. Thus, the total charge , in dollars, for hours of work can be represented as , where is the slope and is the y-intercept of the graph of the equation in the xy-plane. Since the given graph represents the total charge, in dollars, by an electrician for hours of work, it follows that its slope is , or the electrician’s hourly rate.
Choice B is incorrect. The electrician's onetime fee is represented by the y-coordinate of the y-intercept, not the slope, of the graph.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?
Choice B is correct. A system of two linear equations in two variables, and , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, and , are parallel if the coefficients for and in one equation are proportional to the corresponding coefficients in the other equation, meaning ; and the lines are distinct if the constants are not proportional, meaning is not equal to or . The given equation, , can be written in standard form by subtracting from both sides of the equation to yield . Therefore, the given equation can be written in the form , where , , and . The equation in choice B, , is written in the form , where , , and . Therefore, , which can be rewritten as ; , which can be rewritten as ; and , which can be rewritten as . Since , , and is not equal to , it follows that the given equation and the equation are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation has no solution. Thus, the equation in choice B could be the second equation in the system.
Choice A is incorrect. The equation and the given equation represent the same line in the xy-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution.
Choice C is incorrect. The equation and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.
Choice D is incorrect. The equation and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.
Oxygen gas is placed inside a tank with a constant volume. The graph shows the estimated temperature , in kelvins, of the oxygen gas when its pressure is atmospheres.
What is the estimated temperature, in kelvins, of the oxygen gas when its pressure is atmospheres?
Choice C is correct. For the graph shown, the x-axis represents pressure, in atmospheres, and the y-axis represents temperature, in kelvins. Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is atmospheres is represented by the y-coordinate of the point on the graph that has an x-coordinate of . The point on the graph with an x-coordinate of has a y-coordinate of approximately . Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is atmospheres is .
Choice A is incorrect. This is the pressure, in atmospheres, not the estimated temperature, in kelvins, of the oxygen gas when its pressure is atmospheres.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Sadie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?
2
3
4
5
Choice B is correct. Let n be the number of novels and m be the number of magazines that Sadie purchased. If Sadie purchased a total of 11 novels and magazines, then . It is given that the combined price of 11 novels and magazines is $20. Since each novel sells for $4 and each magazine sells for $1, it follows that
. So the system of equations below must hold.
Subtracting corresponding sides of the second equation from the first equation yields , so
. Therefore, Sadie purchased 3 novels.
Choice A is incorrect. If 2 novels were purchased, then a total of $8 was spent on novels. That leaves $12 to be spent on magazines, which means that 12 magazines would have been purchased. However, Sadie purchased a total of 11 novels and magazines. Choices C and D are incorrect. If 4 novels were purchased, then a total of $16 was spent on novels. That leaves $4 to be spent on magazines, which means that 4 magazines would have been purchased. By the same logic, if Sadie purchased 5 novels, she would have no money at all ($0) to buy magazines. However, Sadie purchased a total of 11 novels and magazines.
In the given equation, k is a constant. If the equation has infinitely many solutions, what is the value of k ?
The correct answer is 5. Subtracting from both sides of the given equation gives
, so for any value of x,
if and only if
. Therefore, if the given equation has infinitely many solutions, the value of k is 5.
What is the y-coordinate of the y-intercept of the graph of in the xy-plane?
The correct answer is . A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, which is a point with an x-coordinate of . Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Multiplying both sides of this equation by yields . Dividing both sides of this equation by yields . Therefore, the y-coordinate of the y-intercept of the graph of the given equation in the xy-plane is . Note that 189/5 and 37.8 are examples of ways to enter a correct answer.
How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?
The correct answer is 1.5. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied by the percent concentration of the saline solution. This gives ,
, and
, where x is the amount, in liters, of 25% saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation
must be true. Multiplying 3 by 0.10 and distributing 0.15 to
yields
. Subtracting 0.15x and 0.30 from each side of the equation gives
. Dividing each side of the equation by 0.10 yields
. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.
The graph of a system of linear equations is shown. What is the solution to the system?
Choice C is correct. The solution to a system of linear equations is represented by the point that lies on the graph of each equation in the system, or the point where the lines intersect on a graph. On the graph shown, the two lines intersect at the point . Therefore, the solution to the system is .
Choice A is incorrect. This is the y-intercept of the graph of one of the lines shown, not the intersection point of the two lines.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the solution to the system of equations above, what is the value of x ?
The correct answer is 7. Subtracting the second equation from the first equation eliminates the variable y.
Dividing both sides of the resulting equation by 4 yields x = 7.
A science teacher is preparing the 5 stations of a science laboratory. Each station will have either Experiment A materials or Experiment B materials, but not both. Experiment A requires 6 teaspoons of salt, and Experiment B requires 4 teaspoons of salt. If x is the number of stations that will be set up for Experiment A and the remaining stations will be set up for Experiment B, which of the following expressions represents the total number of teaspoons of salt required?
Choice C is correct. It is given that x represents the number of stations that will be set up for Experiment A and that there will be 5 stations total, so it follows that 5 – x is the number of stations that will be set up for Experiment B. It is also given that Experiment A requires 6 teaspoons of salt and that Experiment B requires 4 teaspoons of salt, so the total number of teaspoons of salt required is 6x + 4(5 – x), which simplifies to 2x + 20.
Choices A, B, and D are incorrect and may be the result of not understanding the description of the context.
The function is defined by . What is the value of when ?
Choice D is correct. The value of when can be found by substituting for in the given function. This yields , or . Therefore, when , the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
In triangle , sides and each have a length of centimeters and side has a length of centimeters. The given equation represents this situation. Which of the following is the best interpretation of in this context?
The difference, in centimeters, between the lengths of sides and
The difference, in centimeters, between the lengths of sides and
The sum of the lengths, in centimeters, of the three sides of the triangle
The length, in centimeters, of one of the two sides of equal length
Choice C is correct. It's given that in triangle , sides and each have a length of centimeters. Therefore, the expression represents the sum of the lengths, in centimeters, of sides and . It's also given that side has a length of centimeters. Therefore, the expression represents the sum of the lengths, in centimeters, of sides , , and . Since is the sum of the lengths, in centimeters, of the three sides of the triangle and , it follows that is the sum of the lengths, in centimeters, of the three sides of the triangle.
Choice A is incorrect. The difference, in centimeters, between the lengths of sides and is , not .
Choice B is incorrect. The difference, in centimeters, between the lengths of sides and is , or , not .
Choice D is incorrect. The length, in centimeters, of one of the two sides of equal length is , not .
What is the slope of the graph of in the xy-plane?
The correct answer is . In the xy-plane, the graph of an equation in the form , where and are constants, has a slope of and a y-intercept of . Applying the distributive property to the right-hand side of the given equation yields . Combining like terms yields . This equation is in the form , where and . It follows that the slope of the graph of in the xy-plane is . Note that 55/4 and 13.75 are examples of ways to enter a correct answer.
If , what is the value of
?
2
5
7
12
Choice A is correct. Adding the like terms on the left-hand side of the given equation yields . Dividing both sides of this equation by 5 yields
.
Choice B is incorrect and may result from subtracting 5, not dividing by 5, on both sides of the equation . Choice C is incorrect. This is the value of x, not the value of
. Choice D is incorrect. This is the value of
, not the value of
.
The relationship between two variables, and , is linear. For every increase in the value of by , the value of increases by . When the value of is , the value of is . Which equation represents this relationship?
Choice C is correct. It’s given that the relationship between and is linear. An equation representing a linear relationship can be written in the form , where is the slope and is the y-coordinate of the y-intercept of the graph of the relationship in the xy-plane. It’s given that for every increase in the value of by , the value of increases by . The slope of a line can be expressed as the change in over the change in . Thus, the slope, , of the line representing this relationship can be expressed as , or . Substituting for in the equation yields . It's also given that when the value of is , the value of is . Substituting for and for in the equation yields , or . Subtracting from each side of this equation yields . Substituting for in the equation yields . Therefore, the equation represents this relationship.
Choice A is incorrect. This equation represents a relationship where for every increase in the value of by , the value of increases by , not , and when the value of is , the value of is , not .
Choice B is incorrect. This equation represents a relationship where for every increase in the value of by , the value of increases by , not , and when the value of is , the value of is , not .
Choice D is incorrect. This equation represents a relationship where for every increase in the value of by , the value of increases by , not , and when the value of is , the value of is , not .
Maria plans to rent a boat. The boat rental costs $60 per hour, and she will also have to pay for a water safety course that costs $10. Maria wants to spend no more than $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Maria can rent the boat?
The correct answer is 4. The equation , where h is the number of hours the boat has been rented, can be written to represent the situation. Subtracting 10 from both sides and then dividing by 60 yields
. Since the boat can be rented only for whole numbers of hours, the maximum number of hours for which Maria can rent the boat is 4.
Robert rented a truck to transport materials he purchased from a hardware store. He was charged an initial fee of $20.00 plus an additional $0.70 per mile driven. If the truck was driven 38 miles, what was the total amount Robert was charged?
$46.60
$52.90
$66.90
$86.50
Choice A is correct. It’s given that Robert was charged an initial fee of $20.00 to rent the truck plus an additional $0.70 per mile driven. Let m represent the number of miles the truck was driven. Since the rental charge is $0.70 per mile driven, represents the amount Robert was charged for m miles driven. Let c equal the total amount, in dollars, Robert was charged to rent the truck. The total amount can be represented by the equation
. It’s given that the truck was driven 38 miles, thus
. Substituting 38 into the equation gives
. Multiplying
gives
. Adding these values gives
, so the total amount Robert was charged is $46.60.
Choices B, C, and D are incorrect and may result from setting up the equation incorrectly or from making calculation errors.
A line passes through the points and in the xy-plane. What is the slope of the line?
The correct answer is . For a line that passes through the points and in the xy-plane, the slope of the line can be calculated using the slope formula, . It's given that a line passes through the points and in the xy-plane. Substituting for and for in the slope formula, , yields , or . Therefore, the slope of the line is . Note that 18/11 and 1.636 are examples of ways to enter a correct answer.
What value of is the solution to the given equation?
Choice C is correct. For the given equation, is a factor of both terms on the left-hand side. Therefore, the given equation can be rewritten as , or , which is equivalent to . Multiplying both sides of this equation by yields . Subtracting from both sides of this equation yields .
Choice A is incorrect. This is the value of for which the left-hand side of the given equation equals , not .
Choice B is incorrect. This is the value of for which the left-hand side of the given equation equals , not .
Choice D is incorrect. This is the value of for which the left-hand side of the given equation equals , not .
A discount airline sells a certain number of tickets, x, for a flight for $90 each. It sells the number of remaining tickets, y, for $250 each. For a particular flight, the airline sold 120 tickets and collected a total of $27,600 from the sale of those tickets. Which system of equations represents this relationship between x and y ?
Choice A is correct. The airline sold two types of tickets for this flight: x tickets at $90 each and the remaining tickets, y, at $250 each. Because the airline sold a total of 120 tickets for this flight, it must be true that x + y = 120. The amount, in dollars, collected from the sale of x tickets at $90 each is represented by 90x. The amount, in dollars, collected from the sale of the remaining y tickets at $250 each is represented by 250y. It is given that a total of $27,600 was collected from the sale of all tickets. Therefore, it must also be true that 90x + 250y = 27,600.
Choice B is incorrect. The total number of tickets sold is represented correctly as x + y = 120. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice C is incorrect. The total number of tickets sold has been correctly represented as x + y. However, according to the information given, this total should be equal to 120, not 27,600, as shown in choice C. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice D is incorrect. The two equations given in choice D have no meaning in this context.
For the given function , the graph of in the xy-plane is parallel to line . What is the slope of line ?
The correct answer is . It’s given that function is defined by . Therefore, the equation representing the graph of in the xy-plane is , and the graph is a line. For a linear equation in the form , represents the slope of the line. Since the value of in the equation is , the slope of the line defined by function is . It's given that line is parallel to the line defined by function . The slopes of parallel lines are equal. Therefore, the slope of line is also .
What is the y-intercept of the line graphed?
Choice D is correct. The y-intercept of a line graphed in the xy-plane is the point where the line intersects the y-axis. The line graphed intersects the y-axis at the point . Therefore, the y-intercept of the line graphed is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The equation gives the relationship between the side lengths and of a certain parallelogram. If , what is the value of ?
The correct answer is . It's given that the equation gives the relationship between the side lengths and of a certain parallelogram. Substituting for in the given equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, if , the value of is .
A movie theater charges $11 for each full-price ticket and $8.25 for each reduced-price ticket. For one movie showing, the theater sold a total of 214 full-price and reduced-price tickets for $2,145. Which of the following systems of equations could be used to determine the number of full-price tickets, f, and the number of reduced-price tickets, r, sold?
Choice B is correct. The movie theater sells f full-price tickets and r reduced-price tickets, so the total number of tickets sold is f + r. Since the movie theater sold a total of 214 full-price and reduced-price tickets for one movie showing, it follows that f + r = 214. The movie theater charges $11 for each full-price ticket; thus, the sales for full-price tickets, in dollars, is given by 11f. The movie theater charges $8.25 for each reduced-price ticket; thus, the sales for reduced-price tickets, in dollars, is given by 8.25r. Therefore, the total sales, in dollars, for the movie showing is given by 11f + 8.25r. Since the total sales for all full-price and reduced-price tickets is $2,145, it follows that 11f + 8.25r = 2,145.
Choice A is incorrect. This system of equations suggests that the movie theater sold a total of 2,145 full-price and reduced-price tickets for a total of $214. Choice C is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket. Choice D is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket and sold a total of 2,145 tickets for a total of $214.
The function f is defined by , where m and b are constants. If
and
, what is the value of m ?
The correct answer is 2. The slope-intercept form of an equation for a line is , where m is the slope and b is the y-coordinate of the y-intercept. Two ordered pairs,
and
, can be used to compute the slope using the formula
. It’s given that
and
; therefore, the two ordered pairs for this line are
and
. Substituting these values for
and
gives
, or 2.
A line in the xy-plane has a slope of and passes through the point . The equation defines the line, where and are constants. What is the value of ?
The correct answer is . It’s given that the equation defines the line. In this equation, represents the slope of the line and represents the y-coordinate of the y-intercept of the line. It’s given that the line has a slope of . Therefore, the value of is .
The population of snow leopards in a certain area can be modeled by the function P defined above, where is the population t years after 1990. Of the following, which is the best interpretation of the equation
?
The snow leopard population in this area is predicted to be 30 in the year 2020.
The snow leopard population in this area is predicted to be 30 in the year 2030.
The snow leopard population in this area is predicted to be 550 in the year 2020.
The snow leopard population in this area is predicted to be 550 in the year 2030.
Choice C is correct. It’s given that represents the population of snow leopards t years after 1990.
corresponds to
and
. It follows that
corresponds to 30 years after 1990, or 2020. Thus, the best interpretation of
is that the snow leopard population in this area is predicted to be 550 in the year 2020.
Choices A and B are incorrect and may result from reversing the interpretations of t and . Choice D is incorrect and may result from determining that 30 years after 1990 is 2030, not 2020.
For groups of or more people, a museum charges per person for the first people and for each additional person. Which function gives the total charge, in dollars, for a tour group with people, where ?
Choice A is correct. A tour group with people, where , can be split into two subgroups: the first people and the additional people. Since the museum charges per person for the first people and for each additional person, the charge for the first people is and the charge for the additional people is . Therefore, the total charge, in dollars, is given by the function , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph shows a linear relationship between and . Which equation represents this relationship, where is a positive constant?
Choice C is correct. The equation representing the linear relationship shown can be written in slope-intercept form , where is the slope and is the y-intercept of the line. The line shown passes through the points and . Given two points on a line, and , the slope of the line can be calculated using the equation . Substituting and for and , respectively, in this equation yields , which is equivalent to , or . Since is the y-intercept, it follows that . Substituting for and for in the equation yields . Adding to both sides of this equation yields . Multiplying this equation by yields . It follows that the equation , where is a positive constant, represents this relationship.
Choice A is incorrect. The graph of this relationship passes through the point , not .
Choice B is incorrect. The graph of this relationship passes through the point , not .
Choice D is incorrect. The graph of this relationship passes through the point , not .
What value of satisfies the equation ?
Choice B is correct. Subtracting from both sides of the given equation yields . Dividing both sides of this equation by yields . Therefore, the value of that satisfies the given equation is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of that satisfies the equation , not .
Choice D is incorrect. This is the value of that satisfies the equation , not .
If , what is the value of when ?
The correct answer is . Substituting for in the given equation yields , or . Therefore, the value of is when .
The cost of renting a carpet cleaner is for the first day and for each additional day. Which of the following functions gives the cost , in dollars, of renting the carpet cleaner for days, where is a positive integer?
Choice A is correct. It's given that the cost of renting a carpet cleaner is for the first day and for each additional day. Therefore, the cost , in dollars, of renting the carpet cleaner for days is the sum of the cost for the first day, , and the cost for the additional days, . It follows that , which is equivalent to , or .
Choice B is incorrect. This function gives the cost of renting a carpet cleaner for days if the cost is , not , for the first day and for each additional day.
Choice C is incorrect. This function gives the cost of renting a carpet cleaner for days if the cost is , not , for the first day and , not , for each additional day.
Choice D is incorrect. This function gives the cost of renting a carpet cleaner for days if the cost is , not , for the first day and , not , for each additional day.
Naomi bought both rabbit snails and nerite snails for a total of . Each rabbit snail costs and each nerite snail costs . If Naomi bought nerite snails, how many rabbit snails did she buy?
Choice A is correct. Let represent the number of rabbit snails that Naomi bought. It’s given that each rabbit snail costs . Therefore, the total cost, in dollars, of the rabbit snails that Naomi bought can be represented by the expression . It’s also given that each nerite snail costs , and that Naomi bought nerite snails. Therefore, the total cost, in dollars, of the nerite snails that Naomi bought is , or . Since Naomi bought both the rabbit snails and the nerite snails for a total of , the equation can be used to represent the situation. Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, Naomi bought rabbit snails.
Choice B is incorrect. This is the total cost, in dollars, of the nerite snails that Naomi bought, not the number of rabbit snails.
Choice C is incorrect. This is the cost, in dollars, of one rabbit snail and one nerite snail, not the number of rabbit snails that Naomi bought.
Choice D is incorrect and may result from conceptual or calculation errors.
The shaded region shown represents the solutions to which inequality?
Choice D is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form , where is the slope and is the y-intercept of the line. For the graph shown, the boundary line passes through the points and . Given two points on a line, and , the slope of the line can be calculated using the equation . Substituting the points and for and in this equation yields , which is equivalent to , or . Since the point represents the y-intercept, it follows that . Substituting for and for in the equation yields as the equation of the boundary line. Since the shaded region represents all the points above this boundary line, it follows that the shaded region shown represents the solutions to the inequality .
Choice A is incorrect. This inequality represents a region below, not above, a boundary line with a slope of , not .
Choice B is incorrect. This inequality represents a region below, not above, the boundary line shown.
Choice C is incorrect. This inequality represents a region whose boundary line has a slope of , not .
In the xy-plane, line passes through the point and is parallel to the line represented by the equation . If line also passes through the point , what is the value of ?
Choice C is correct. A line in the xy-plane can be represented by an equation of the form , where is the slope and is the y-coordinate of the y-intercept of the line. It's given that line passes through the point . Therefore, the y-coordinate of the y-intercept of line is . It's also given that line is parallel to the line represented by the equation . Since parallel lines have the same slope, it follows that the slope of line is . Therefore, line can be represented by the equation , where and . Substituting for and for in yields the equation , or . If line passes through the point , then when , for the equation . Substituting for and for in this equation yields , or .
Choice A is incorrect. This is the y-coordinate of the y-intercept of the line represented by the equation .
Choice B is incorrect. This is the slope of the line represented by the equation .
Choice D is incorrect. The line represented by the equation , not line , passes through the point .
If , what is the value of ?
Choice C is correct. It’s given that . Multiplying each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Last week, an interior designer earned a total of from consulting for hours and drawing up plans for hours. The equation represents this situation. Which of the following is the best interpretation of in this context?
The interior designer earned per hour consulting last week.
The interior designer worked hours drawing up plans last week.
The interior designer earned per hour drawing up plans last week.
The interior designer worked hours consulting last week.
Choice A is correct. It's given that represents the situation where an interior designer earned a total of last week from consulting for hours and drawing up plans for hours. Thus, represents the amount earned, in dollars, from consulting for hours, and represents the amount earned, in dollars, from drawing up plans for hours. Since represents the amount earned, in dollars, from consulting for hours, it follows that the interior designer earned per hour consulting last week.
Choice B is incorrect. The interior designer worked hours, not hours, drawing up plans last week.
Choice C is incorrect. The interior designer earned per hour, not per hour, drawing up plans last week.
Choice D is incorrect. The interior designer worked hours, not hours, consulting last week.
The solution to the given system of equations is . What is the value of ?
The correct answer is . It's given by the second equation in the system that . Substituting for in the first equation in the system, , yields . Subtracting from both sides of this equation yields .
Angela is playing a video game. In this game, players can score points only by collecting coins and stars. Each coin is worth c points, and each star is worth s points.
Which system of equations can be used to correctly determine the values of c and s ?
Choice B is correct. The number of coins collected can be multiplied by c to give the score from the points earned from coins. Similarly, the number of stars collected can be multiplied by s to give the score from the points earned from the stars. Therefore, the total score each time Angela played is , and the total score the second time she played is
.
Choices A, C, and D are incorrect and may result from misidentifying the terms of the equation. Choice A switches coins and stars, choice C switches stars and points, and choice D misidentifies coins, stars, and points.
The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population and year is linear, which of the following functions P models the population of Greenleaf t years after 2000?
Choice A is correct. It is given that the relationship between population and year is linear; therefore, the function that models the population t years after 2000 is of the form , where m is the slope and b is the population when
. In the year 2000,
. Therefore,
. The slope is given by
. Therefore,
, which is equivalent to the equation in choice A.
Choice B is incorrect and may be the result of incorrectly calculating the slope as just the change in the value of P. Choice C is incorrect and may be the result of the same error as in choice B, in addition to incorrectly using t to represent the year, instead of the number of years after 2000. Choice D is incorrect and may be the result of incorrectly using t to represent the year instead of the number of years after 2000.
A linear function gives a company’s profit, in dollars, for selling items. The company’s profit is when it sells items, and its profit is when it sells items. Which equation defines ?
Choice D is correct. It’s given that the relationship between and is linear. A linear function can be written in the form , where is the slope and is the y-coordinate of the y-intercept of the graph of in the xy-plane. Given two points on a line, and , the slope of the line can be found using the slope formula . It’s given that the company’s profit is when it sells items and the profit is when it sells items. Since represents the company's profit, in dollars, for selling items, the graph of in the xy-plane passes through the points and . Substituting and for and , respectively, in the slope formula yields , which gives , or . Substituting for , for , and for in yields , or . Subtracting from each side of this equation yields . Substituting for and for in yields . Therefore, the equation that defines is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
Choice D is correct. The value of can be found by isolating this expression in the given equation. Subtracting from both sides of the given equation yields . Subtracting from both sides of this equation yields , which gives , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
The y-intercept of the graph of in the xy-plane is . What is the value of ?
The correct answer is . It’s given that the y-intercept of the graph of is . Substituting for in this equation yields , or . Therefore, the value of that corresponds to the y-intercept of the graph of in the xy-plane is .
If , what is the value of ?
Choice D is correct. It’s given that . Multiplying both sides of this equation by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
What is the equation of the line shown in the xy-plane above?
Choice B is correct. Any line in the xy-plane can be defined by an equation in the form y = mx + b, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. From the graph, the y-intercept of the line is (0, 3). Therefore, b = 3. The slope of the line is the change in the value of y divided by the change in the value of x for any two points on the line. The line shown passes through (0, 3) and (1, 0), so , or m = –3. Therefore, the equation of the line is y = –3x + 3.
Choices A and C are incorrect because the equations given in these choices represent a line with a positive slope. However, the line shown has a negative slope. Choice D is incorrect because the equation given in this choice represents a line with slope of . However, the line shown has a slope of –3.
What is the solution to the equation ?
1
1.5
2
4
Choice C is correct. Subtracting 3 from both sides of the given equation yields . Dividing both sides by 2 results in
.
Choices A and B are incorrect and may result from computational errors. Choice D is incorrect. This is the value of .
For the linear function , is a constant. When , . What is the value of ?
Choice D is correct. It’s given that when , . Substituting for and for in the given function yields , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If a new graph of three linear equations is created using the system of equations shown and the equation , how many solutions will the resulting system of three equations have?
Zero
Exactly one
Exactly two
Infinitely many
Choice A is correct. A solution to a system of equations must satisfy each equation in the system. It follows that if an ordered pair is a solution to the system, the point lies on the graph in the xy-plane of each equation in the system. The only point that lies on each graph of the system of two linear equations shown is their intersection point . It follows that if a new graph of three linear equations is created using the system of equations shown and the graph of , this system has either zero solutions or one solution, the point . Substituting for and for in the equation yields , or . Since this equation is not true, the point does not lie on the graph of . Therefore, is not a solution to the system of three equations. It follows that there are zero solutions to this system.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Davio bought some potatoes and celery. The potatoes cost per pound, and the celery cost per pound. If Davio spent in total and bought twice as many pounds of celery as pounds of potatoes, how many pounds of celery did Davio buy?
Choice D is correct. Let represent the number of pounds of potatoes and let represent the number of pounds of celery that Davio bought. It’s given that potatoes cost per pound and celery costs per pound. If Davio spent in total, then the equation represents this situation. It’s also given that Davio bought twice as many pounds of celery as pounds of potatoes; therefore, . Substituting for in the equation yields , which is equivalent to , or . Dividing both sides of this equation by yields . Substituting for in the equation yields , or . Therefore, Davio bought pounds of celery.
Choice A is incorrect. This is the number of pounds of potatoes, not the number of pounds of celery, Davio bought.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of when ?
Choice D is correct. The value of when can be found by substituting for in the given equation . This yields , or . Therefore, when , the value of is .
Choice A is incorrect. This is the value of when , rather than the value of when .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
Choice C is correct. It's given that . Adding to both sides of this equation yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
A candle is made of ounces of wax. When the candle is burning, the amount of wax in the candle decreases by ounce every hours. If ounces of wax remain in this candle, for how many hours has it been burning?
Choice D is correct. It’s given that the candle starts with ounces of wax and has ounces of wax remaining after a period of time has passed. The amount of wax the candle has lost during the time period can be found by subtracting the remaining amount of wax from the amount of wax the candle was made of, which yields ounces, or ounces. This means the candle loses ounces of wax during that period of time. It’s given that the amount of wax decreases by ounce every hours. If represents the number of hours the candle has been burning, it follows that . Multiplying both sides of this equation by yields . Therefore, the candle has been burning for hours.
Choice A is incorrect and may result from using the equation rather than to represent the situation, and then rounding to the nearest whole number.
Choice B is incorrect. This is the amount of wax, in ounces, remaining in the candle, not the number of hours it has been burning.
Choice C is incorrect and may result from using the equation rather than to represent the situation.
In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a ?
4
9
36
54
Choice C is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side of the equation and a constant on the right-hand side of the equation. The coefficients of x and y in the second equation are equal to the coefficients of x and y, respectively, in the first equation multiplied by 4: and
. Therefore, the constant in the second equation must be equal to 4 times the constant in the first equation:
, or
.
Choices A, B, and D are incorrect. When ,
, or
, the given system of equations has no solution.
Monarch butterflies can fly only with a body temperature of at least . If a monarch butterfly's body temperature is , what is the minimum increase needed in its body temperature, in , so that it can fly?
Choice B is correct. It's given that monarch butterflies can fly only with a body temperature of at least . Let represent the minimum increase needed in the monarch butterfly's body temperature to fly. If the monarch butterfly's body temperature is , the inequality represents this situation. Subtracting from both sides of this inequality yields . Therefore, if the monarch butterfly's body temperature is , the minimum increase needed in its body temperature, in , so that it can fly is .
Choice A is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is , not .
Choice C is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is , not .
Choice D is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is , not .
The table shows two values of and their corresponding values of . In the xy-plane, the graph of the linear equation representing this relationship passes through the point . What is the value of ?
Choice D is correct. The linear relationship between and can be represented by the equation , where is the slope of the graph of this equation in the xy-plane and is the y-coordinate of the y-intercept. The slope of a line between any two points and on the line can be calculated using the slope formula . Based on the table, the graph contains the points and . Substituting and for and , respectively, in the slope formula yields , which is equivalent to , or . Substituting for , for , and for in the equation yields , or . Adding to both sides of this equation yields . Therefore, and . Substituting for and for in the equation yields . Thus, the equation represents the linear relationship between and . It's also given that the graph of the linear equation representing this relationship in the xy-plane passes through the point . Substituting for and for in the equation yields , which is equivalent to , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A cleaning service that cleans both offices and homes can clean at most places per day. Which inequality represents this situation, where is the number of offices and is the number of homes?
Choice A is correct. It's given that the cleaning service cleans both offices and homes, where is the number of offices and is the number of homes the cleaning service can clean per day. Therefore, the expression represents the number of places the cleaning service can clean per day. It's also given that the cleaning service can clean at most places per day. Since represents the number of places the cleaning service can clean per day and the service can clean at most places per day, it follows that the inequality represents this situation.
Choice B is incorrect. This inequality represents a cleaning service that cleans at least places per day.
Choice C is incorrect. This inequality represents a cleaning service that cleans at most more offices than homes per day.
Choice D is incorrect. This inequality represents a cleaning service that cleans at least more offices than homes per day.
A store sells two different-sized containers of blueberries. The store’s sales of these blueberries totaled dollars last month. The equation represents this situation, where is the number of smaller containers sold and is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container?
The correct answer is . It’s given that the equation represents this situation, where is the number of smaller containers sold, is the number of larger containers sold, and is the store’s total sales, in dollars, of blueberries last month. Therefore, represents the store's sales, in dollars, of smaller containers, and represents the store's sales, in dollars, of larger containers. Since is the number of smaller containers sold, the price, in dollars, of each smaller container is .
The function gives the total number of people on a company retreat with managers. What is the total number of people on a company retreat with managers?
The correct answer is . It's given that the function gives the total number of people on a company retreat with managers. It's also given that managers are on the company retreat. Substituting for in the given function yields , or . Therefore, there are a total of people on a company retreat with managers.
Which of the following is the graph of the equation in the xy-plane?
Choice D is correct. In the xy-plane, the graph of the equation , where m and b are constants, is a line with slope m and y-intercept
. Therefore, the graph of
in the xy-plane is a line with slope 2 and a y-intercept
. Having a slope of 2 means that for each increase in x by 1, the value of y increases by 2. Only the graph in choice D has a slope of 2 and crosses the y-axis at
. Therefore, the graph shown in choice D must be the correct answer.
Choices A, B, and C are incorrect. The graph of in the xy-plane is a line with slope 2 and a y-intercept at
. The graph in choice A crosses the y-axis at the point
, not
, and it has a slope of
, not 2. The graph in choice B crosses the y-axis at
; however, the slope of this line is
, not 2. The graph in choice C has a slope of 2; however, the graph crosses the y-axis at
, not
.
Which of the following ordered pairs (x, y) satisfies the system of inequalities above?
Choice D is correct. Any point (x, y) that is a solution to the given system of inequalities must satisfy both inequalities in the system. The second inequality in the system can be rewritten as . Of the given answer choices, only choice D satisfies this inequality, because inequality
is a true statement. The point
also satisfies the first inequality.
Alternate approach: Substituting into the first inequality gives
, or
, which is a true statement. Substituting
into the second inequality gives
, or
, which is a true statement. Therefore, since
satisfies both inequalities, it is a solution to the system.
Choice A is incorrect because substituting −2 for x and −1 for y in the first inequality gives , or
, which is false. Choice B is incorrect because substituting −1 for x and 3 for y in the first inequality gives
, or
, which is false. Choice C is incorrect because substituting 1 for x and 5 for y in the first inequality gives
, or
, which is false.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Adding the first equation to the second equation in the given system yields , or . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields . Therefore, the value of is .